Preheat oven to 350F. Lightly grease the bottom and sides of a 9-inch round springform pan.
In a large bowl, mix 1 3/4 cups flour, 1 1/2 teaspoons baking powder and 1/4 teaspoon salt. Set aside.
In a medium bowl, toss 1 pound apples with 2 tablespoons fresh orange juice. Stir in 1/3 cup brown sugar, 1 1/2 teaspoons cardamom and 1 teaspoon cinnamon. Set aside.
Using a handheld electric mixer or a stand mixer fitted with the paddle attachment, beat 1/2 cup butter, 1 8-ounce block cream cheese, 1 cup sugar and 1 teaspoon vanilla until pale and fluffy (about 3 minutes on medium-high speed). Add the eggs one at a time, beating well after each addition. Scrape the bottom and sides of your bowl.
Switch to low speed and add the flour in 3 stages. Beat until smooth and well combined. Again remember to scrape your bowl.
Use a rubber spatula to gently fold in your apples until evenly distributed. Transfer the batter to your springform pan and spread evenly.
For the crumble, mix all ingredients in a small bowl until well incorporated. Sprinkle on top of your batter. Bake for 60 minutes or until golden brown and the edges start to separate from the pan. A toothpick inserted in the middle should come out with minimal crumbs.
Cool in the pan for 10 minutes. Run a sharp knife around the edges of the pan before releasing the sides. Serve warm.