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. Toss gently to coat the peaches evenly. Let the mixture sit for 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 just combined, being careful not to overmix.

Pour the peach mixture into a greased 9x13-inch baking dish. Drop spoonfuls of the biscuit dough evenly over the top of the peaches. It's okay if the dough doesn't cover the entire surface; the cobbler will expand as it bakes.

Place the baking dish in the preheated oven and bake 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, cover the dish loosely with aluminum foil.

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