Preheat oven to 375°F (190°C).
In a bowl, mix peaches with granulated sugar, brown sugar, cinnamon, nutmeg, vanilla, lemon juice, and cornstarch until well coated.
Spread peach mixture evenly into a buttered 9×13-inch baking dish.
In another bowl, whisk flour, sugar, baking powder, and salt.
Add milk and whisk until smooth, then stir in melted butter.
Pour batter evenly over the peaches without stirring.
Bake for 40–45 minutes until golden brown and bubbly.
Let rest 10–15 minutes before serving.