Preheat the oven to 350ºF. Line a 9x9 inch baking pan with parchment paper and spray with nonstick cooking spray.

In a large bowl, whisk together the wet ingredients until well combined: melted butter, applesauce, eggs, maple syrup, and vanilla extract.

In a separate medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, ginger, allspice, nutmeg, and salt. Add the dry ingredients to the wet ingredients and mix until just combined; do not overmix. Fold in the chopped apples.

Pour the batter into the prepared pan and smooth the top with a rubber spatula. Bake for 30-35 minutes or until a tester comes out clean. Allow the cake to cool completely before frosting.

While the cake cooks and cools, brown the butter for the frosting: Melt the butter in a small saucepan over medium heat. Once melted, whisk constantly; the butter will begin to crackle, then foam. After a few minutes, the butter will begin to turn a golden amber color. As soon as the butter turns brown and gives off a nutty aroma, 5 to 8 minutes total, pour the brown butter into a small bowl and transfer to the fridge for 45 to 60 minutes to allow the butter to resolidify so that it’s similar to room temperature butter but not completely hardened (poking it should leave an indent). If it hardens too much, allow it to soften up at room temperature.

In the bowl of a stand mixer fitted with the whisk attachment (or in a large mixing bowl if using a hand mixer), beat the cooled brown butter on high speed until light and fluffy, about 1 minute. Add the cream cheese and beat on high until creamy, about 1 minute. Add the powdered sugar, vanilla, and salt and beat until light and fluffy, 2 to 3 minutes.

Spread the frosting over the cooled cake. Sprinkle a light dusting of cinnamon and a big pinch of flaky sea salt on top, then cut the cake into 16 slices and enjoy! Leftover cake should be stored covered in the fridge for up to 5 days.