1. Prepare the yeast mixture: Combine the warm milk, yeast, and 1 tablespoon of the sugar in a bowl. Stir gently and leave for about 8 to 10 minutes until foamy and active. If using fresh yeast, crumble it into the milk first and stir until dissolved.

2. Mix the dough ingredients: In a large mixing bowl, combine the flour, remaining sugar, and salt. Add the yeast mixture, egg yolks, whole egg, sour cream, vanilla, and melted butter. Mix until a rough dough forms and no dry flour remains.

3. Knead the dough: Knead the dough for about 8 to 10 minutes using a stand mixer with a dough hook or by hand for 12 to 15 minutes. The dough should become smooth, elastic, and slightly tacky but not sticky. Add only a small amount of extra flour if absolutely necessary.

4. First rise: Transfer the dough to a lightly greased bowl. Cover and let rise in a warm place for about 60 to 75 minutes or until doubled in size.

5. Make the apple filling: While the dough rises, place the diced apples, sugar, lemon juice, cinnamon, and vanilla in a saucepan over medium heat. Cook for 8 to 10 minutes, stirring often, until the apples soften but still hold some shape. Stir in the cornstarch during the final minute of cooking. Remove from heat and cool completely.

6. Prepare the crumble topping: Place the flour and sugar in a bowl. Add the cold butter and rub everything together with your fingertips until small crumbs form. Refrigerate until needed.

7. Shape the buns: Line two baking sheets with parchment paper. Transfer the risen dough to a lightly floured surface and divide it into 12 equal pieces. Shape each piece into a smooth ball and place on the baking sheets, leaving enough space between each bun.

8. Create the centers: Gently flatten each dough ball into a thick round about 4 inches (10 cm) wide. Using the bottom of a glass or your fingertips, press a deep well into the center of each bun while leaving a raised border around the edges.

9. Fill the buns: Spoon a generous amount of cooled apple filling into the center of each bun. Do not overfill, as the filling may spill over during baking.

10. Add the crumble: Sprinkle the prepared crumble evenly over the apple filling. Some crumble can also fall slightly onto the dough edges for a classic bakery appearance.

11. Second rise: Cover the shaped buns loosely with a clean towel and let them rise for another 20 to 25 minutes until puffy.

12. Apply the egg wash: Whisk together the egg yolk and milk. Brush only the exposed dough edges, avoiding the apple filling and crumble.

13. Bake: Bake in a preheated 350°F (180°C) oven for 18 to 22 minutes or until the buns are deeply golden around the edges and fully baked through. Rotate the baking sheets halfway through baking if needed for even coloring.

14. Cool completely: Transfer the buns to a cooling rack and allow them to cool for at least 20 minutes before icing.

15. Make the icing: Whisk together the powdered sugar, milk, and vanilla until smooth. Adjust with a few drops of milk if needed to create a thick but pourable consistency.

16. Finish the buns: Drizzle the icing over the cooled buns in thin ribbons. Allow the icing to set for about 10 minutes before serving.