Preheat the oven to 375°F (190°C). Grease a 9x13-inch baking dish and set aside.
In a large bowl, combine the sliced peaches, 0.5 cup granulated sugar, brown sugar, lemon juice, cornstarch, cinnamon, nutmeg and a pinch of salt. Toss gently to coat and let sit for 10 minutes to release juices.
Pour the melted butter into the prepared baking dish, spreading it evenly across the bottom.
In a separate bowl, whisk together the flour, 0.25 cup granulated sugar, baking powder and salt. Add the milk, beaten egg and vanilla and stir until just combined into a thick batter.
Spoon the batter by large tablespoons over the melted butter in the dish; it will spread during baking. Arrange the peach mixture evenly over the batter. Pour any remaining peach juices over the top.
Bake in the preheated oven for 40 to 45 minutes, or until the topping is golden brown and peaches are bubbly. If the top browns too quickly, tent loosely with foil.
Remove from the oven and let cool for 10 to 15 minutes before serving. Serve warm with vanilla ice cream or whipped cream if desired.
Notes
Use the ripest peaches you can find for best flavor. The cobbler can be reheated and is excellent with a scoop of vanilla ice cream.