First we will begin making the dough. Heat up milk in a small sauce pan until it begins to simmer. Remove from heat and add sugar and salt and stir. Let it cool down to lukewarm to the touch. Once milk is cool, add to a bowl of a stand mixer, along with 3 ½ cups of the flour, melted butter, egg, egg yolks, and yeast. Attach the dough hook to the stand mixer and mix on low until dough forms. Increase speed to medium and knead dough until it's smooth and no longer sticks to the sides, about 10 minutes. If the dough is sticking for a while, add more dough until it only sticks to the bottom.
Place a couple tablespoons of flour on the surface of your counter. Place dough on top of it and begin kneading the dough. Keep kneading for about 30 seconds until the dough is smooth and elastic. Once you finish kneading, shape into a ball. Lightly grease another large bowl and place the ball of dough into bowl. Cover with a kitchen towel and let it rise for 1.5 hours.
Prepare two baking sheets by lining with parchment paper. Once dough has risen, punch it down with your fist and remove from the bowl. Lightly dust your counter with flour and place the dough on top. Form into a disk. Cut dough into 4 even slices. Cut each slice into 4 even pieces. You should have 16 even pieces. Roll each piece of dough into a ball and place on a baking sheet with parchment paper. Make sure to space the balls a couple inches apart and only 8 balls per baking sheet. Cover each baking sheet with a damp rag and let them rise until fluffy, about 1.5 hours.
While the dough is rising, prepare the apples. Place 2 tablespoons of butter into a large skillet and over medium heat . Once it's melted, add diced apples and cook until slightly softened, about 8 minutes. Turn off heat. Add sugar, cinnamon, nutmeg, and salt. Stir together until combined. Set aside.
Prepare the streusel. Add 3 tablespoons each of flour and sugar to a small bowl. Add 1 tablespoon of cold diced butter to mixture and mix together with your fingers until the mixture resembles sand. Place in refrigerator to keep cold.
Preheat oven to 350 degrees. Make sure both oven racks are placed in the center of the oven. Once the balls have puffed up, remove the rag that covers them. Grab a 1/3-cup measuring cup and dip into flour. Then use it to smash the center of each ball of dough to create a well in the center. Redip the cup in flour if it starts to stick to the dough. Fill each well with about 3 tablespoons of cooked apples.
Brush the exposed edges of the kolaches with egg wash. To make the egg wash, whisk egg with 1 tablespoon of water. Sprinkle with chilled streusel.
Bake both sheets of kolaches in preheated oven until they are golden brown on the edges, about 25 minutes. Make sure to switch and rotate baking sheets halfway to ensure even baking.
Remove from oven and let cool for 15 minutes. Then, make icing by mixing powdered sugar and cream together. Drizzle over kolaches. Serve immediately or within 24 hours.