Preheat oven to 350 degrees.: The warm scent of a properly preheated oven is an early sign that the dish will bake evenly, producing uniform doneness across the pan. You should feel a steady heat if you hover your hand briefly near the open door, and the oven should already have recovered its temperature before you slide the pan in. A common mistake is putting the dish into a cold oven, which causes uneven set and longer bake times, so wait until the thermostat shows a steady 350.
In a small saucepan bring half & half and butter to a boil. Add salt and stir.: As the half & half and melted butter warm, you will see small bubbles form at the edges and the surface will shimmer, releasing a gentle dairy aroma. Stirring helps the fat emulsify into the liquid for a smooth sauce. Watch carefully to avoid a vigorous boil, which can scald the dairy and create a skin. If you notice foaming or a scorched smell, reduce the heat immediately and skim any solids.
Place hash browns in a 9×13 inch pan. Pour 3 c. of half & half mixture over hash browns. Sprinkle Parmesan cheese on top. Bake in preheated oven for 50 minutes.: Arrange the shredded potato patties evenly so there are no dense clumps, which ensures even cooking and consistent texture. The visual cue you want is a uniform layer across the pan without gaps, which helps the liquid distribute properly. A common slip is overcrowding or leaving large voids, both of which lead to uneven set and textural differences between portions.
Remove from and pour the rest of the half and half mixture over the hash browns and bake 15 minutes longer.: When the warm liquid meets the shredded potato , you should see an immediate darkening as the starch hydrates, and the surface will glisten from the melted butter . This initial pour saturates the layer so the center becomes creamy while the top still has room to brown. If you pour all the liquid at once you risk a soggy top, so reserve the remainder as the recipe indicates to control the final texture.
Sprinkle Parmesan cheese on top.: As you scatter the grated Parmesan cheese , notice how the dry shreds sit atop the wet surface, ready to toast. Once heated, the cheese will melt and create small golden patches that provide savory contrast. Avoid piling the cheese unevenly, because dense clusters can burn; aim for a balanced distribution for even browning.
Bake in preheated oven for 50 minutes.: During this long bake, the kitchen will fill with a warm, toasty dairy scent and you will see the edges begin to bubble gently and the top attain a light golden tone. That bubbling is a good signal that the interior is setting. A typical error here is opening the oven too often, which lets heat escape and elongates the bake. Check through the oven window if possible and trust the timer.
Remove from and pour the rest of the half and half mixture over the hash browns and bake 15 minutes longer.: After the initial bake, pouring the reserved warm mixture revitalizes the interior moisture and encourages a creamier mouthfeel as it finishes. You will notice a renewed gloss on the surface and a subtle steam rise when it goes back into the heat, which helps form a tender center while the top finishes browning. Be careful when handling the hot pan, and avoid pouring too quickly which can cause splashes and pockets of unmixed liquid; pour steadily and spread any pooled liquid with a spatula if needed.