In a skillet over medium-low heat, melt the butter. Add the diced apples, brown sugar, lemon juice, and cinnamon. Cook for 8–10 minutes, stirring frequently, until the apples are nearly tender.

In a small bowl, combine the water and cornstarch to form a slurry. Stir the slurry into the apple mixture and cook for an additional 1–2 minutes, or until the juices have thickened. Remove from heat, stir in the vanilla, and transfer to a bowl. Refrigerate until completely cooled.

In a small saucepan, whisk together the water and bread flour until smooth. Cook over medium heat, stirring continuously, until the mixture thickens into a paste and reaches approximately 150°F (66°C). Remove from heat and transfer to a bowl to cool for about 20 minutes.

Warm the milk to 95°F (35°C). Stir in the active dry yeast and brown sugar. Allow the mixture to rest for 5–7 minutes, or until foamy and activated.

In the bowl of a stand mixer, combine the bread flour, cooled tangzhong, brown sugar, melted butter, egg, salt, and activated yeast mixture. Mix until a shaggy dough forms.

Using the dough hook attachment, knead on medium-low speed for 10–15 minutes, or until the dough becomes smooth and elastic. If necessary, add additional flour one tablespoon at a time; the dough should remain slightly tacky but not overly sticky.

Form the dough into a smooth ball and place it in a lightly greased bowl. Cover and allow to rise until doubled in size, approximately 45 minutes to 1 ½ hours, depending on room temperature.

In a medium bowl, beat the cream cheese until smooth. Add the sour cream, powdered sugar, lemon zest, and vanilla. Mix until fully incorporated and smooth. Refrigerate until ready to use.

In a small bowl, combine the flour, brown sugar, granulated sugar, and cinnamon. Add the cold cubed butter and work it into the mixture using your fingertips until coarse crumbs form. Refrigerate until needed.

Line two baking sheets with parchment paper. Once the dough has doubled in size, divide it into 8 equal portions (approximately 85g each). Shape each portion into a smooth ball and place on the prepared baking sheets.

Cover loosely and allow the buns to proof for about 20 minutes, until slightly puffed. Preheat the oven to 350°F (180°C). Prepare the egg wash by whisking together the egg and water.

Using the base of a small jar or measuring cup, create a well in the center of each bun. Press gently with your fingers to deepen the pocket if needed.

Brush the outer edges lightly with egg wash. Spoon the cheesecake filling into each well, followed by approximately 2 heaping tablespoons of the apple filling. Sprinkle streusel over the tops and around the edges.

Bake for 20–25 minutes, rotating the pans halfway through baking. The buns are ready when golden brown and the internal temperature reaches approximately 190°F (88°C).

Allow to cool slightly before serving. Best enjoyed warm.