Preheat your oven to 375°F (190°C). In a large bowl, combine the sliced peaches, granulated sugar, brown sugar, cinnamon, nutmeg, vanilla extract, and lemon juice. Stir until the peaches are evenly coated. Let the mixture sit for 10 minutes to allow the juices to release.

In a separate bowl, mix the flour, baking powder, and salt. Add the cold cubed butter and use a pastry cutter or your fingers to work the butter into the flour mixture until it resembles coarse crumbs. Gradually add the milk, stirring until the dough comes together.

Pour the peach mixture into a greased 9x13-inch baking dish. Drop spoonfuls of the cobbler topping evenly over the peaches. Spread the topping slightly to cover some of the peaches, but don't worry about covering them completely.

Bake in the preheated oven for 40-45 minutes, or until the topping is golden brown and the peach filling is bubbly. If the topping begins to brown too quickly, cover the dish with foil for the remaining baking time.

Allow the cobbler to cool for a few minutes before serving. It's best enjoyed warm, perhaps with a scoop of vanilla ice cream or a dollop of whipped cream.