Preheat oven to 450 degrees F (230 degrees C).

Whisk 3/4 cup brown sugar, 1 1/2 tablespoons flour, and salt together in a large bowl; whisk in sour cream, egg, and vanilla extract. Stir in apples; turn apple mixture into unbaked pie crust.

Bake in the preheated oven for 10 minutes. Remove pie from oven; reduce oven temperature to 350 degrees F (175 degrees C).

Place 1/2 cup flour, 1/2 cup brown sugar, cinnamon, and nutmeg in a bowl. Cut in butter until mixture resembles coarse crumbs; sprinkle crumble topping evenly over pie.

Bake pie until apples are tender, about 30 minutes. Cool before serving, about 30 minutes.