Prep: Butter the bottom and sides of a 9 x 13 casserole dish.: The air warms with a hint of butter when you rub the bottom and sides of the dish, which helps prevent sticking and encourages an even edge crust; use softened butter to coat thoroughly with your fingers or a paper towel. You'll feel the sheen on the pan, and the subtle scent of dairy signals the pan is ready. Avoid leaving big globs of butter pooled in corners, which can cause localized browning; spread it evenly for best results.
Using a mandolin, thinly slice potatoes into rounds about 1/8" thick. Place potato slices in a bowl with cold water while you prep the sauce. This keeps them from discoloring and removes some of the starch. Drain potatoes well, just before you put them in your casserole dish.: When you slide the potatoes across a mandolin, they should produce clean, translucent rounds that shimmer slightly. Submerge them in cold water immediately to stop oxidation and wash away surface starch, which helps prevent gluey layers. Drain just before layering so they are not waterlogged; too wet and the sauce will be thin, too dry and they will absorb more liquid and tighten. A common mistake is slicing unevenly, which leads to uneven cooking; keep a steady motion and consistent thickness.
Preheat Oven to 400˚F. In a medium saucepan over medium/high heat, melt 4 Tbsp butter. Add finely diced yellow onion and sauté until soft and golden (5 min). Add pressed garlic cloves and sauté another 1-2 min, stirring constantly, until fragrant. Whisk in 2 Tbsp flour and cook another 1 min until flour is golden. Add 1 1/2 cups milk and 1 cup chicken broth. Whisk mixture until smooth and bring to a simmer. Stir in 1/2 Tbsp sea salt and 1/2 tsp pepper (or season to taste). Remove from heat.: You will notice warmth filling the oven as it reaches temperature, which sets the stage for steady baking. Preheating ensures the gratin begins cooking immediately and helps the sauce come up to temperature with the potatoes . If you skip full preheat, the dish may take longer and the texture can become gummy rather than tender.
Drain potatoes and arrange half of them in the bottom of your casserole dish (it doesn't matter how the first layer is arranged, just separate the potato slices as you put them in). Pour half of your sauce over the top and sprinkle with 1/4 cup parmesan cheese. Repeat with remaining potatoes, sauce and top with 1/4 cup parmesan cheese. I made the second layer of potatoes prettier for presentation.: As the butter melts, it will sizzle softly and release a nutty scent; keep the heat steady so it foams then calms without burning. This step forms the flavor base, and maintaining medium/high heat helps the next aromatics to sweat rather than brown quickly. Watch closely because butter can brown in moments at high heat, which will change the sauce flavor.
Cover with aluminum foil and bake in the center of the oven at 400˚F for 55 minutes then uncover and broil another 3-5 minutes or until the top is golden brown. Garnish with green onion if desired and serve.: The diced onion will become translucent then develop sweet, golden edges as natural sugars caramelize. Stir frequently so pieces cook evenly; you want softness and a touch of color, not deep browning. Overcooking to a dark brown will lend a roasted rather than sweet character, which alters the intended flavor balance.
Add pressed garlic cloves and sauté another 1-2 min stirring constantly until fragrant: Once the garlic hits the warm butter and onion , it will release a quick, heady aroma indicating readiness. Stir constantly to prevent small bits from burning, which creates bitterness. The garlic should smell aromatic and sweet when done, not acrid; if it darkens, start the sauce step again or reduce heat next time.
Whisk in 2 Tbsp flour and cook another 1 min until flour is golden: As you whisk the flour into the fat, the mixture will thicken into a smooth paste that loses its raw taste after about a minute. You should see the roux take on a pale golden hue and smell slightly nutty. This step is critical because the roux is what thickens the sauce, so undercooking leaves a pasty flavor, while overcooking will darken it too much and reduce thickening power.
Add 1 1/2 cups milk and 1 cup chicken broth: Pouring in the liquid will produce a few hissy bubbles as the roux integrates; whisk vigorously to dissolve lumps and create a glossy sauce. The mixture should become silkier and begin to thicken on gentle simmer. If lumps form, strain or whisk harder until smooth. A thin sauce at this stage means either not enough roux or too much liquid; simmer to reduce, or add a touch more roux dissolved in cold liquid.
Whisk mixture until smooth and bring to a simmer: When the sauce begins to ripple and small bubbles form at the edges, the starches are activating and the texture will thicken. You will smell a mild dairy fragrance and see the sauce coat a spoon. Keep it at a gentle simmer to avoid scorching the bottom, and stir often to maintain even heat. Boiling aggressively can separate the sauce or scald the milk, leaving an off taste.
Stir in 1/2 Tbsp sea salt and 1/2 tsp pepper (or season to taste): Adding the sea salt and black pepper at this point seasons the sauce properly, lifting flavors of the potatoes and parmesan cheese . Taste carefully because the chicken broth and parmesan cheese also contribute salt. Over seasoning is common here, so season lightly and adjust after baking if needed.
Remove from heat: The sauce should be smooth, warm, and slightly thickened when you take it off the heat, losing the small bubbles and becoming glossy. Removing it prevents it from reducing too much and allows for easier layering. Keep in mind residual heat will continue gentle thickening, so do not overreduce on the stove.
Drain potatoes and arrange half of them in the bottom of your casserole dish: As you layer, the potatoes will feel cool and slightly starchy to the touch; space them so edges overlap but are not crowded, allowing the sauce to seep between slices. This layering helps even cooking and a coherent structure. Crowding slices causes inconsistent doneness and dense texture, so aim for a loose, single layer.
Pour half of your sauce over the top and sprinkle with 1/4 cup parmesan cheese: The sauce should pool gently around each slice, creating pockets of creaminess, while the parmesan cheese begins to melt into the surface and add savory depth. Use a spoon to coax sauce into gaps so every slice is coated. A common slip is pouring sauce unevenly, leaving dry pockets, which results in inconsistent texture.
Repeat with remaining potatoes sauce and top with 1/4 cup parmesan cheese: The second layer should mirror the first, and finishing with parmesan cheese on top creates the attractive browned crust after broiling. Make the top layer neater for presentation if you like, aligning edges for a tidy look. If the top is too dry before baking, brush a little extra sauce on it to encourage browning rather than burning.
Cover with aluminum foil and bake in the center of the oven at 400˚F for 55 minutes: Covered baking steams the interior, producing tender potatoes , while keeping moisture in so the sauce infuses the layers. After the time elapses, the center should feel just set and the aroma will be thick and savory. Avoid removing the foil early, which can dry the interior; only uncover later for browning.
Uncover and broil another 3-5 minutes or until the top is golden brown: Under the broiler you will hear a light crackle, and the parmesan cheese will form a golden, slightly crisp crust with toasted notes. Keep the dish a few inches from the heat source and watch constantly, since broiling can go from golden to burned in seconds. If the top darkens too quickly, lower the rack or reduce broil time.
Garnish with green onion if desired and serve: A sprinkle of optional green onions adds fresh color and a gentle onion bite against the rich, creamy backdrop. Let the gratin rest a few minutes so the sauce sets slightly, making it easier to serve neat slices. Serving too hot may cause the sauce to run and lose shape, so a brief rest yields cleaner portions.