Prepare the dough: Whisk the milk, yeast, and 1 tablespoon of sugar together in the bowl of your stand mixer fitted with a dough hook or paddle attachment. Cover and allow to sit for 5 minutes.
Add the remaining sugar, the butter, egg yolks, salt, and 1 cup flour. Beat on low speed for 30 seconds, scrape down the sides of the bowl with a rubber spatula, then add another cup of flour. Beat on medium speed until relatively incorporated. Scrape down the sides of the bowl with a rubber spatula. Add 1/2 cup of flour and beat on medium speed until the dough begins to come together. As the mixer runs, add another 2-4 tablespoons of flour depending on how wet the dough looks. This should be a very soft and almost creamy-feeling dough. Do not add more flour than you need.
Knead the dough: Keep the dough in the mixer and beat for an additional 3 minutes or knead by hand on a lightly floured work surface for 3 minutes.
1st Rise: Lightly grease a large bowl with nonstick spray. Place the dough in the bowl, turning it to coat all sides. Cover the bowl and allow the dough to rise in a relatively warm environment for 3-4 hours or until nearly double in size. This dough is rich with fat, so it takes longer than other doughs to rise.
Prepare the filling: Melt 1 tablespoon of butter in a large skillet on the stove over medium heat. Add the apples. Stir and cook until the apples are slightly softened, about 3 minutes. Remove from heat and stir in lemon juice to help prevent browning. Set aside. For the rest of the filling in this step, make sure you don't prepare it too far in advance because the butter will solidify and spreading onto the dough will be difficult. Melt remaining 2 tablespoons of butter in a medium heatproof bowl in the microwave (or use the stove). Stir in brown sugar, flour, cinnamon, salt, and vanilla extract until crumbly and combined. Stir in the egg white. Set aside.
Grease a 9×5 inch loaf pan.
Shape the dough: After the dough rises, punch it down to release the air. Flour a work surface, your hands, and a rolling pin. Roll the dough out into a 12x16 inch rectangle. Carefully and slowly spread the cinnamon filling mixture on top. Add the apples in a single layer. Using floured hands, tightly roll up the dough to form a 16-inch long log. Place the log on its seam. Fold in half, then twist it to form a figure 8. Pinch the ends together.
2nd Rise: Cover shaped babka and allow to rise until it's puffy and nearly reaches the top of the loaf pan, about 1-1½ hours.
Prepare the crumble topping: Mix the brown sugar, flour, and cinnamon together. Add the cold butter and using a pastry cutter or a fork, cut butter into the brown sugar mixture until small crumbles form. Refrigerate until ready to use.
Preheat oven to 350°F. Place a baking sheet on a lower oven rack to catch any juices or crumbles that may drip/drop down.
Carefully brush the surface of the babka with last remaining egg white. Using a toothpick, poke 10-12 holes all over the top of the loaf. Sprinkle crumble topping on top.
Bake: Bake for 60-65 minutes or until golden brown on top. If you gently tap on the loaf, it should sound hollow. If you notice the top browning too quickly, loosely tent the pan with aluminum foil. Remove from the oven and allow bread to cool in the pan for at least 30 minutes before slicing and serving. For extra neat slices, I recommend cooling completely before slicing.