Preheat your oven to 375°F (190°C).

In a large mixing bowl, combine the sliced apples, 1/2 cup granulated sugar, 2 tbsp all-purpose flour, ground cinnamon, ground nutmeg, lemon juice, 1 tsp vanilla extract, and melted butter. Toss gently until the apples are evenly coated.

Pour the apple mixture into a 9x13 inch baking dish and spread it evenly across the bottom.

In a separate large mixing bowl, whisk together 1 1/2 cups all-purpose flour, 1/2 cup granulated sugar, baking powder, and salt.

Add the cold, cubed butter to the dry ingredients. Use a pastry blender, two knives, or your fingertips to cut the butter into the flour mixture until it resembles coarse crumbs.

Pour in the chilled milk and 1/2 tsp vanilla extract. Stir with a spoon or spatula until just combined. Be careful not to overmix; the dough will be soft and slightly sticky.

Drop spoonfuls of the cobbler topping over the apple mixture in the baking dish. You don't need to cover the entire surface completely; leaving some gaps will allow the apple filling to bubble through.

If desired, sprinkle the coarse sugar over the topping for a beautiful sparkle and extra crunch.

Bake for 40-45 minutes, or until the apple filling is bubbly around the edges and the cobbler topping is golden brown and cooked through. To check for doneness, a knife inserted into the topping should come out clean.

Remove from the oven and let the cobbler cool for at least 15-20 minutes before serving. This allows the filling to set slightly.