In a large mixing bowl, combine the all-purpose flour, warm whole milk, egg, instant yeast, granulated sugar, vanilla sugar, and salt. Stir until a shaggy dough forms, then knead on a floured surface for about 5-10 minutes until smooth and elastic.

Incorporate the softened unsalted butter and continue kneading until you achieve a soft, pliable dough that springs back when poked.

Place the dough in a lightly greased bowl, covering it with a clean kitchen towel. Allow it to rise in a warm, draft-free area until doubled in size, which should take around 1 hour.

While the dough rises, peel and dice the apples into small pieces. In a skillet, sauté the apples with brown sugar and cinnamon over medium heat for about 5-7 minutes until they soften and caramelize.

Combine cornstarch with a little water, then stir it into the apple mixture and cook until thickened. Let the filling cool completely before using.

In a medium bowl, whisk the cream cheese with granulated sugar and vanilla sugar until it achieves a smooth and creamy consistency.

Once the dough has risen, gently punch it down to release any air bubbles. Divide the dough into 9 equal portions and shape each piece into a ball. Allow the dough balls to rest for about 15 minutes on a floured surface.

Using a floured glass or your fingers, press the centers of each dough ball to create a well.

Lightly brush the edges of each indentation with the egg yolk to help seal the filling. Spoon a layer of the cheesecake mixture into each well, followed by a generous scoop of the apple mixture.

In a small bowl, combine cold butter, granulated sugar, and flour for the crumble topping. Use your fingers or a pastry cutter to mix until crumbly.

Preheat your oven to 180°C (350°F). Arrange the filled buns on a parchment-lined baking sheet, spacing them an inch apart.

Bake for 20-25 minutes, or until they are golden brown and puffed up.

Once cooled slightly, whisk together powdered sugar and milk in a small bowl until you achieve a drizzling consistency. Drizzle the glaze over the warm buns.