Combine Wet Ingredients: In a large bowl, whisk together the eggs, Greek yogurt, melted butter, granulated sugar, and vanilla sugar until smooth and well combined, about 1 to 2 minutes.
Add Leavening Agents: Mix in a small portion of the flour, then add the baking soda and baking powder, stirring gently to incorporate.
Incorporate Flour: Gradually add the remaining flour and mix until just combined, being careful not to overmix, which can result in a dense cake.
Spiced Apples: Toss the diced apples with vanilla sugar and cinnamon in a separate bowl until evenly coated, about 1 minute.
Crumb Mixture: In another bowl, combine the flour, sugar, and cold butter, then rub together with your fingertips until coarse crumbs form, about 2 to 3 minutes.
Mix Glaze Base: In a medium bowl, beat together the cream cheese, powdered sugar, vanilla sugar, and cinnamon until smooth and creamy, about 1 minute.
Adjust Glaze: Gradually add the milk, one tablespoon at a time, until the glaze reaches your desired consistency for drizzling, about 2 to 3 minutes.
Prepare Pan: Preheat the oven to 350°F (180°C) and line a 9x13-inch baking pan with parchment paper for easy removal.
Layer Batter: Spread most of the batter into the prepared pan, reserving enough for a thin top layer to seal in the apples.
Add Apples: Distribute the cinnamon apples evenly over the batter.
Top and Crumb: Spread the remaining batter over the apples, then sprinkle the crumb topping evenly over the entire surface.
Bake: Bake for 45 to 50 minutes, until the top is golden brown and a toothpick inserted into the center comes out clean or with just a few moist crumbs.
Cool: Let the cake cool in the pan for at least 20 to 30 minutes before turning out onto a wire rack.
Drizzle: Once completely cooled, drizzle the cream cheese glaze over the top of the cake in a decorative pattern.
