Preheat the oven to 350°F (175°C).Toss the sliced peaches with both sugars, lemon juice, vanilla, cinnamon, nutmeg, and cornstarch.Spread the peach mixture evenly into a greased baking dish.In another bowl, whisk together the flour, sugar, baking powder, salt, milk, and melted butter until smooth.Pour the batter evenly over the peaches without stirring.Bake for 45–50 minutes until the topping is golden brown and the filling is bubbling.Let the cobbler cool for 15 minutes before serving.Serve warm with vanilla ice cream or whipped cream.