Prep the oven and pan: Heat the oven to 350°F (175°C). Place the melted butter in a 9x13-inch baking dish and swirl to coat the bottom.
Sweeten the peaches: In a bowl, toss peaches with 1/2 cup granulated sugar, brown sugar, cinnamon, nutmeg, and lemon juice. Let sit 10 minutes to get nice and juicy.
Make the batter: In another bowl, whisk flour, 1/2 cup sugar, baking powder, and salt.
Stir in milk and vanilla until smooth. The batter will be pourable.
Add batter to the pan: Pour the batter evenly over the melted butter. Do not stir. The butter helps create that signature crust.
Add the peaches: Spoon the peaches and their juices evenly over the batter. Do not mix. The batter will rise around the fruit as it bakes.
Bake: Place in the oven and bake 40–50 minutes, until the top is golden brown and set, and the peach juices are bubbling around the edges.
Rest and serve: Let the cobbler cool 10–15 minutes so it thickens slightly. Serve warm with vanilla ice cream or whipped cream.