Preheat your oven to 375°F. Place the pie crust into a 9-inch pie plate, crimp the edges, and refrigerate while you prepare the filling.
In a large mixing bowl, combine the sliced apples, granulated sugar, brown sugar, flour, cinnamon, nutmeg, salt, lemon juice, and vanilla extract. Toss until the apples are evenly coated.
Pour the apple mixture into the prepared pie crust, gently pressing the apples down so they're evenly distributed.
In another bowl, mix together the flour, brown sugar, cinnamon, and salt for the crumb topping. Cut in the cold butter using a pastry cutter or your fingertips until the mixture resembles coarse crumbs.
Sprinkle the crumb topping evenly over the apples, covering the entire pie.
Place the pie on a baking sheet to catch any drips and bake for 50–60 minutes, or until the topping is golden brown and the filling is bubbling around the edges.
If the crumb topping begins to brown too quickly, loosely cover the pie with aluminum foil during the last 15–20 minutes of baking.
Remove the pie from the oven and allow it to cool for at least 2 hours before slicing. This helps the filling set for clean slices.