Preheat the Oven:
Preheat your oven to 350°F (175°C). Grease a 9-inch baking dish or casserole dish lightly with butter.
Prepare the Peach Filling:
In a large bowl, combine the sliced peaches, sugar, lemon juice, cornstarch, cinnamon, and nutmeg. Stir until well-mixed and the peaches are evenly coated. Pour the peach mixture into the prepared baking dish.
Make the Cobbler Topping:
In a separate bowl, whisk together the flour, baking powder, and salt. Add the melted butter, milk, and vanilla extract, and stir until a thick batter forms.
Assemble the Cobbler:
Drop spoonfuls of the cobbler batter over the peach filling. Do not worry about covering the entire surface; the batter will spread and puff up as it bakes.
Bake:
Bake in the preheated oven for 40-45 minutes or until the topping is golden brown and the peach filling is bubbling around the edges.
Cool and Serve:
Allow the peach cobbler to cool for about 10 minutes before serving. For added indulgence, serve with vanilla ice cream or whipped cream on top.