Position an oven rack in the lower third of the oven and heat to 375°F (190°C). Place a rimmed baking sheet on the rack below where the baking dish will sit and set out a 9x13-inch (23x33 cm) baking dish.
Peel the 8 apples, cut them in half, and remove the cores. Slice each half into ¼-inch (6 mm) pieces and place in a large mixing bowl. Pour the lemon juice over them right away to slow browning.
Whisk the brown sugar, granulated sugar, cornstarch, cinnamon, nutmeg, and salt in a small bowl until no cornstarch lumps remain. Sprinkle over the apples, add the vanilla, and toss until every slice is coated. Let sit for 10 minutes.
Pour the apples and all their juices into the baking dish, spread into an even layer, and press down lightly to remove large gaps. Scatter the 2 tablespoons of butter pieces over the top.
In a medium bowl, whisk the flour, ¼ cup granulated sugar, baking powder, and ½ teaspoon salt until evenly combined.
Add the cold butter cubes and cut them in with a pastry cutter until the mixture looks like coarse crumbs with some pea-sized pieces still visible. Keep the butter cold and do not overwork it.
Pour in the cold milk and stir with a rubber spatula just until the dough comes together and no dry flour remains. Drop mounds of dough (about 2 tablespoons each) over the apples, leaving ½-inch (1.25 cm) gaps between them. Mix the turbinado sugar and ½ teaspoon cinnamon, then sprinkle over the mounds.
Bake for 40 to 45 minutes, until the biscuits are deep golden brown and the filling bubbles thickly at the edges and in the center. Tent with foil for the last 10 minutes if the topping browns too fast.
Move the dish to a wire cooling rack and let it rest for 20 to 30 minutes so the juices thicken. Serve warm, spooning apples and biscuit into each bowl.