Preheat the oven to 350 degrees F. Grease an 8×8-inch baking pan with cooking spray and set aside.: You will notice the kitchen warming and the air taking on a subtle roasted note as the oven reaches temperature, which primes the pan to receive the batter evenly. Preheating ensures the bake begins cooking at an even rate, producing edges that set while the center finishes without overbaking. A common pitfall is skipping the greasing step, which can cause sticking, so make sure the pan is well prepared. If your oven runs hot, place the rack in the center to prevent the top from browning too quickly.
Place the butter, in tablespoon pieces, in a medium saucepan and melt over medium heat, stirring often, until it foams, smells nutty, and browns. This will take about 5 to 7 minutes. Scrape the browned butter into a small bowl. Let cool for 5 to 10 minutes.: You will hear gentle bubbling, then see the foam subside and amber flecks appear as proteins caramelize, releasing a nutty, warm aroma that signals the step is done. This scent is transformative, giving depth to the final dish. Letting it cool slightly prevents cooking the eggs when combined later. A common mistake is walking away and burning the butter, which leaves a bitter note, so stay attentive and stir constantly. If you smell anything acrid, discard and start with fresh butter.
In a large bowl, combine the oats, brown sugar, baking powder, cinnamon, and salt. Set aside.: The dry mixture will smell faintly of spice and molasses, and combining these components ensures even distribution of leavening and seasoning so each bite is balanced. You want the oats coated with the dry ingredients so they absorb the liquid uniformly. Avoid over mixing here, or you can compress the oats and make the texture heavier than intended. If your baking powder is old, the bake will be flatter, so check freshness.
Pour the milk over the oat mixture. Add the browned butter, eggs, maple syrup, and vanilla extract. Stir until well combined. Fold in the chopped peaches.: At this stage you will see the batter come together into a cohesive mass, with the milk hydrating the oats and the browned butter blending into glossy ribbons. The aroma will deepen as the maple and vanilla mingle. Folding the chopped peaches gently preserves their shape so they create juicy pockets instead of dissolving. A common issue is over stirring, which can break the fruit apart and lead to excess moisture; fold carefully and stop when even distribution is achieved.
Pour the mixture into the prepared pan. Top with sliced peaches and sprinkle with turbinado sugar.: As you spread the batter, it should hold its shape, not run like a custard. Arrange the sliced peaches on top in a single layer for an attractive finish, and the turbinado sugar will catch the heat to form tiny, crunchy highlights. This visual layer also signals where servings will be most fruit forward. Be careful not to press the fruit into the batter, which could sink them; place them gently.
Bake for 35 to 40 minutes or until the oats are set and lightly golden brown. Remove from the oven and let cool in the pan for 5 minutes. Cut into squares and serve with whipped cream, if desired.: The oven will do the heavy lifting, transforming loose batter into a structured yet tender bake. Look for a golden top and a slightly firm center that springs back when touched; an inserted toothpick should come out mostly clean with a few moist crumbs. Cooling briefly helps the squares hold together for neat slices. A frequent error is underbaking, which leaves a soggy center, so if the center seems very soft, give it another 5 minutes and recheck. Serve warm, and add whipped cream for extra richness if you like.