Preheat and Prepare: Preheat your oven to 375°F (190°C). Grease a 9x13 inch baking dish with butter or cooking spray.
Prepare the Peaches: If using fresh peaches, bring a large pot of water to boil. Score an X on the bottom of each peach and blanch for 30-60 seconds. Transfer to ice water, then peel off the skins. Slice peaches into 1/2-inch thick wedges.
Make Peach Filling: In a large bowl, combine sliced peaches, granulated sugar, brown sugar, cornstarch, vanilla extract, cinnamon, nutmeg, and salt. Toss gently until peaches are evenly coated. Let stand for 10 minutes to allow juices to develop.
Transfer Filling: Pour the peach mixture into the prepared baking dish and dot with butter pieces. Bake for 15 minutes while you prepare the topping.
Make Cobbler Topping: In a medium bowl, whisk together flour, sugar, baking powder, and salt. Cut in cold butter using a pastry cutter or two forks until mixture resembles coarse crumbs with some pea-sized pieces.
Combine Wet Ingredients: In a separate small bowl, whisk together milk, egg, and vanilla extract. Pour into the flour mixture and stir just until combined. Don't overmix.
Assemble: Remove the partially baked peaches from the oven. Drop spoonfuls of the cobbler topping evenly over the hot peaches, leaving some gaps for the fruit to show through.
Final Bake: Return to the oven and bake for 30-35 minutes, or until the topping is golden brown and a toothpick inserted into the topping comes out clean.
Cool and Serve: Allow to cool for 10-15 minutes before serving. Serve warm with vanilla ice cream or whipped cream if desired.