Preheat the oven to 350ºF (180ºC). Melt the butter in a skillet, watching it carefully and swirling it gently. When it starts to take on a light brown color and smells a little nutty, and the bubbles begin to subside, pour the browned butter into an 8-inch (20 cm) square baking pan, leaving the blackened bits back in the skillet.: You will notice a toasty, nutty aroma and small brown flecks forming in the butter , which signals the Maillard reaction adding depth. The sizzling sound will quiet as the water evaporates; that is your cue to act. Pouring the hot fat into the pan creates a base that the batter will contact and eventually bake into a tender crust, so do this step carefully to avoid splatters. If you burn the butter , it will smell acrid and impart bitterness, so remove it from heat as soon as the scent turns pleasantly nutty. Use an oven mitt and tilt the skillet slowly to control the flow, leaving the darker solids behind so they do not overwhelm the dish.
In a medium bowl mix together the flour, sugar, baking powder, lemon zest, and salt with a whisk. Add the milk and vanilla extract, and stir until smooth.: As you whisk dry ingredients like flour and baking powder , you should see a uniform pale mixture with flecks of lemon zest scattered through. Adding the milk and vanilla extract transforms this into a glossy batter; it should pour slowly but not be runny. Stirring until smooth eliminates lumps, but do not overwork the batter or it may develop a chewy texture. If you see pockets of unincorporated flour, fold gently until integrated. A common mistake is over mixing which makes the topping dense, so stop when streaks are gone and the batter looks cohesive.
Pour the milk mixture over the butter in the pan without stirring. (Don't worry, the batter will rise up over the peaches as they are baking.) Toss the peach slices with the cinnamon and strew them over the batter in an even layer. Crumble the brown sugar over the top and bake until the top is golden brown and the center of the cobbler feels just set, about 40 minutes. Remove from the oven and cool on a wire rack.: When the batter hits the hot butter , you will hear a faint hiss and see a ripple as the two layers settle. Leaving them separate allows the batter to bake into pockets that rise up around the fruit, producing a light, tender crumb. Resist the urge to mix; stirring collapses those layers and changes the texture. If you accidentally stir, the cobbler will still bake but will be more uniform and less airy, so pour gently and level the surface with a light shake if needed.
Toss the peach slices with the cinnamon and strew them over the batter in an even layer: As you mix the sliced peaches with ground cinnamon , the spice will perfume the fruit and release a warm scent. Arrange the slices in a single even layer so heat reaches each piece uniformly; overlapping can create uneven cooking where some parts become mushy while others remain underdone. The peaches will sink and the batter will rise around them, creating that classic cobbler texture. Avoid piling slices in the center, instead distribute them to encourage even bubbling.
Crumble the brown sugar over the top and bake until the top is golden brown and the center of the cobbler feels just set, about 40 minutes. Remove from the oven and cool on a wire rack: As the cobbler bakes you will see bubbling juices around the edges and a golden top forming where the light brown sugar caramelizes. The scent will shift to warm, toffee notes with peach perfume. Testing the center should reveal a gentle jiggle but not a liquid center; that is how you know it is set. Overbaking dries the topping and underbaking leaves a raw batter taste, so aim for a golden color and a barely set middle. Let it cool slightly on a wire rack so the juices thicken and you can serve without losing all the sauce to the plate.