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 about 10 minutes to allow the juices to form.

In a separate bowl, whisk together 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. Stir in the milk 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. Don't worry about covering every inch; the topping will spread as it bakes.

Bake in the preheated oven for 35-40 minutes, or until the topping is golden brown and the peach filling is bubbly. If the topping begins to brown too quickly, tent the dish with aluminum foil.

Let the cobbler cool for about 10 minutes before serving. This allows the juices to thicken slightly. Serve warm with a scoop of vanilla ice cream or a dollop of whipped cream for an extra treat.