Preheat your oven to 350°F Grease a 9×13-inch baking pan or line it with parchment paper. Set aside.
In a medium bowl, mix together the brown sugar, granulated sugar, cinnamon, and salt. Stir in the melted butter until combined. Add the flour and gently mix with a fork until crumbly. Be careful not to overmix; you want clumps.
3/4 cup packed light brown sugar, 1/3 cup granulated sugar, 1 teaspoon ground cinnamon, 1/2 teaspoon sea salt, 3/4 cup unsalted butter, 2 cups all-purpose flour
In a separate bowl, whisk together the flour, baking soda, baking powder, and salt. Set aside.
2 and 1/3 cups all-purpose flour, 1 teaspoon baking soda, 1 teaspoon baking powder, 3/4 teaspoon salt
Using a handheld or stand mixer fitted with the paddle attachment, beat the softened butter and granulated sugar together on high speed until light and creamy, about 2 minutes. Scrape down the sides of the bowl as needed.
3/4 cup unsalted butter, 1 and 1/4 cups granulated sugar
Add the eggs, sour cream, applesauce, and vanilla to the creamed butter and sugar. Beat on medium-high speed until everything is well combined. The mixture may look curdled—this is normal. Scrape down the sides and bottom of the bowl as needed.
4 large eggs, 3/4 cup full-fat sour cream, 1/4 cup applesauce, 2 teaspoons pure vanilla extract
With the mixer on low speed, gradually add the dry ingredients to the wet mixture, mixing until smooth. Do not overmix—the batter should be thick.
In a small bowl, toss the chopped apples with cinnamon until evenly coated.
2 cups peeled and chopped Granny Smith apples, 1/2 teaspoon ground cinnamon
Spread the cake batter evenly into the prepared baking pan. Evenly distribute the cinnamon-coated apples on top of the batter. Finally, sprinkle the crumb topping over the apples, pressing it down lightly with the back of a spoon so it adheres to the batter.
Bake for 45-55 minutes, or until a toothpick inserted into the center comes out clean. If the top or edges begin browning too quickly, cover the cake loosely with aluminum foil.
Once baked, remove the cake from the oven and place it on a wire rack. Let it cool for at least 30-45 minutes before slicing and serving.