Preheat the oven to 350°F (175°C).
Prepare the crumb topping by mixing 1/2 cup flour, 1/2 cup brown sugar, and 1/2 teaspoon cinnamon in a bowl. Add 1/4 cup melted butter and mix until crumbly.
Prepare the apple filling by mixing sliced apples, 2 tablespoons granulated sugar, and 1 teaspoon cinnamon in a bowl.
Mix the cake batter by combining 2 1/4 cups flour, 1 1/2 cups granulated sugar, 1/2 teaspoon baking powder, and 1/4 teaspoon salt in a large bowl.
Add 1/2 cup melted butter, 2 large eggs, 1 teaspoon vanilla extract, and 3/4 cup sour cream to the dry ingredients. Mix until well combined.
Assemble the cake by spreading half of the cake batter in a 9-inch deep-dish pie pan, followed by the apple filling, and then the remaining cake batter.
Top the cake with the crumb topping and bake for 50-60 minutes or until a toothpick inserted in the center comes out clean.
Let the cake cool for 10 minutes before serving.