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. Mix well and let it sit for 10 minutes to allow the sugars to dissolve and the peaches to release their juices.

In another bowl, whisk together the flour, baking powder, and salt. Add the chilled butter cubes 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 biscuit dough evenly over the peaches. It's okay if the dough doesn't cover the entire surface; the cobbler will expand 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. Let it cool for a few minutes before serving.