Prepare the brown butter: In a small saucepan, melt ½ cup of butter over medium heat. Continue to cook, swirling occasionally, until the butter turns golden brown and releases a nutty aroma (about 5-7 minutes). Remove from heat and let it cool.

Mix dry ingredients: In a large bowl, whisk together the flour, baking powder, baking soda, apple pie spice, and sea salt. Set aside.

Combine wet ingredients: In another bowl, whisk together the brown sugar, granulated sugar, oil, eggs, and vanilla extract. Add the apple butter and grated apple, mixing until combined.

Add brown butter: Slowly stir in the cooled brown butter into the wet ingredients. Once everything is well incorporated, fold in the dry ingredients until just combined. Be careful not to overmix!

Bake the cake: Pour the batter into a greased 9-inch round cake pan, spreading it evenly. Bake at 350°F (175°C) for 45 to 50 minutes, or until a toothpick inserted into the center comes out clean. Allow the cake to cool in the pan for 10 minutes before transferring it to a wire rack to cool completely.

Make the frosting: While the cake is cooling, melt the remaining ½ cup of butter in a small saucepan over medium heat until it is browned. Remove the saucepan from the heat and let the butter cool slightly. In a mixing bowl, combine the browned butter with powdered sugar, vanilla extract, and a pinch of sea salt. Stir the mixture until it is smooth and creamy.

Frost the cake: Once the cake is fully cooled, spread the brown butter frosting over the top. Slice, serve, and enjoy the perfect bite of fall!