Prepare the peaches by peeling and slicing them. Combine with 1/2 cup of sugar and lemon juice, toss gently, and let sit for 10 minutes.
Make the cobbler batter by whisking together flour, baking powder, salt, 1/2 cup sugar, cinnamon, and nutmeg. Add milk, melted butter, and vanilla, and stir until just combined.
Pour the batter into a greased 9x13-inch baking dish. Spoon the marinated peaches and juice over the batter without stirring.
Preheat the oven to 350°F. Bake for 40-45 minutes until golden brown and a toothpick comes out clean. Let cool for 10 minutes before serving.