Prepare the Dough: Whisk the warm milk, yeast, and 1 tablespoon of the sugar in the bowl of your stand mixer. Cover and let this sit for 5-10 minutes until foamy on the top. (If you do not own a stand mixer, you can do this in a large mixing bowl and in the next step, mix the dough together with a large wooden spoon.)
¾ cup whole milk, 2¼ teaspoons active dry yeast
Add the remaining sugar, melted butter, eggs, salt, and bread flour. With a dough hook mix on low speed for 2-3 minutes, pausing to scrape down the sides as needed to get all the flour mixed in. Then mix on low speed for another 2-3 minutes until the dough comes together and pulls away from the sides of the bowl. Dough should be soft and a little sticky, but it should come away fairly cleanly from the bowl.
⅓ cup granulated sugar, 6 tablespoons unsalted butter, 2 large eggs, ½ teaspoon salt, 3½ cups bread flour
First Rise: Lightly sprinkle no more than 1 tablespoon of flour on a clean working surface. Place the dough on the counter and gently form into a ball. Spray a large bowl with cooking spray and place the dough ball in the prepared bowl, spinning to coat the dough with the oil. Cover and place in a warm place to rise for 1.5-2 hours or until doubled in size.
Prepare a 9x13 baking dish with parchment paper or spray with cooking spray. You can also bake these on a large half sheet baking sheet lined with parchment paper. Shaping: Punch down the dough to release air, and remove the dough to a clean lightly floured surface. Press the dough out into a rectangle shape and divide the dough evenly into 12 portions. Shape each piece into a smooth ball by stretching out the top while pinching the bottom. Place the formed dough into the prepared baking dish.
2nd Rise: Cover the baking dish with a clean kitchen towel and allow to rise until the rolls are puffy, approximately 45 minutes.
Prepare Toppings: While the rolls are in their second rise, prepare the toppings by combining the cream cheese, sugar, egg yolk, and vanilla extract with a pinch of salt. Mix until this is smooth and silky. A hand mixer is helpful for this step. 4 ounces cream cheese, 2 tablespoons sugar, 1 large egg yolk, ½ teaspoon vanilla extract, ⅛ teaspoon salt
Adjust oven rack to center position and preheat to 375℉/190℃.
Fill the Kolaches: Gently press down the center of each kolache roll using a rounded measuring cup, or small shot glass. This may need to be twice if the rolls rise back up after being pressed down. Gently brush the tops of the kolaches with egg wash. Spoon in the jam, don't heap it to overflowing. Then top with a spoonful of the cream cheese mixture. Sprinkle the crumble over the tops.
Bake the Kolaches: Bake for 20-23 minutes or until lightly golden brown on top, rotating the pan halfway if needed for even baking. If you notice the tops browning too much, tent a sheet of aluminum foil over the top. To check that the rolls are fully baked, use an instant read thermometer, the internal temperature should be 190℉ (88℃) Allow the kolaches to cool for several minutes before serving. Store any leftover rolls in a sealed container in the refrigerator for 4-7 days.