Preheat the oven to 350 degrees F, and grease a 9-inch springform pan.
Melt the butter in a medium skillet, over medium heat.
Continue to heat the butter, watching carefully, and allowing it to sizzle/crackle, and turn a warm brown color. Immediately transfer to a small bowl, to halt the cooking process, and allow to cool to room temperature.
Beat the cream cheese together with the browned butter, until creamy.
Mix in the sugar and vanilla extract.
Add the eggs, one at a time.
Finally, mix in the flour, baking powder, and salt, until just combined. Scrape the bottom and sides of the bowl, making sure all of the ingredients are well incorporated.
Pour ⅔ of the batter into the prepared pan, and layer the spiced apples on top. Spread the rest of the batter over the apples, and sprinkle the remaining cinnamon sugar mixture evenly over the top.
Bake for 1 hour and 15 minutes, or until a cake tester inserted in the center of the cake comes out with a few moist crumbs.