Make the apple pie filling. We're starting here, love! Melt 2 tablespoons of butter in a medium saucepan over medium heat, then stir in the brown sugar and cinnamon. Add your diced apples and cook for about 10 minutes, stirring occasionally, until they're just tender. Sprinkle in the cornstarch, stir well, and cook for 1 more minute until thick and glossy. Set it aside to cool.
Brown the butter. Brown all of the remaining butter at once, so you only have to do this step one time! Melt it over medium heat for 5 to 8 minutes, until it smells nutty and the bits at the bottom turn golden. Pour it into a heatproof bowl, scrape in all those brown bits, and let it cool for 10 minutes so it doesn't scramble your eggs.
Get your pan ready. Preheat your oven to 350°F. Line the bottom of a 9-inch springform pan or 9x9-inch pan with parchment paper so the cake pops out super easily.
Mix the wet ingredients. Mix 1/2 cup of brown butter with the brown sugar and granulated sugar for about 2 minutes. Add the eggs and vanilla and mix again. Add the sour cream and milk and mix until completely smooth.
Add the dry ingredients. Whisk together the flour, baking powder, and salt in a separate bowl. Gently fold them into the wet ingredients until everything is just mixed together, then set the batter aside.
Make the crumble. Combine 6 tablespoons of brown butter with the brown sugar, flour, and cinnamon. Use a fork to mix until you have a clumpy crumble. The big clumps are the BEST part!
Layer it up. Spread half of the cake batter into the pan. Sprinkle on 1/3 of the crumble, then add all of the apple pie filling. Gently spread the remaining batter over the top and finish with the remaining crumble.
Bake. Bake at 350°F for 40 to 50 minutes, or until a toothpick comes out clean. Let the cake cool completely. I know it's hard to wait, but the layers need time to set!
Glaze and serve. Whisk together the remaining brown butter, heavy cream, powdered sugar, and vanilla until smooth. You might have a little less than 2 tablespoons of brown butter, and that's totally fine! Drizzle it over the cake, slice into 12 to 16 pieces, and get ready to be OBSESSED.