Make the brown butter: Add 113g of butter to a small saucepan set over medium heat. Stir the butter occasionally until it turns amber in color and develops little brown specks.
Transfer the butter to a medium sized mixing bowl and set it aside to cool while you make the apple pie filling. (You should be left with 90g of butter.)
Apple Filling
Wash, peel, and dice the apples into small cubes.
Add the apples to the saucepan you made the brown butter in, then mix in the sugar and apple pie spice. Add in the butter and set the mixture over medium-heat.
Stir the apples occasionally until the mixture comes to a simmer. Let the apples simmer for 5 minutes, then remove the mixture from the heat and set it aside.
Apple Spiced Sugar
In a small bowl, whisk together the dark brown sugar and apple pie spice. Set aside.
Apple Fritter Cake
Preheat the oven to 350°F / 175°C. Butter a 7" springform pan and coat it with sugar.
To the cooled brown butter, add in the sugar, eggs, vanilla, milk, and sour cream. Whisk until well combined.
Add in the flour, baking powder, and salt. Mix just until the last streak of flour disappears.
Get a small mixing bowl and place a fine mesh sieve over it. Pour the apples into the fine mesh sieve to remove the extra juice. Set the juice aside to use for the glaze.
Spread half of the batter into your loaf pan, then sprinkle ½ of the apples on top of the batter and ½ of the apple spice sugar over the apples. Carefully spread the rest of the batter over the apples, then top the batter with the remaining apples and the remaining apple spice sugar.
Bake for 40-45 minutes, or until a toothpick inserted in the center of the cake comes out with only a few moist crumbs on it.
Apple Spiced Glaze
Add the powdered sugar to the bowl with the leftover apple-spiced juice in it. Mix until well combined, adding in 1-2 tsp. of milk as needed to get to your desired consistency.
Once the cake has cooled to room temperature, pour the glaze over it, then slice, serve, and enjoy!