Start by making blooming the yeast. Heat whole milk in the microwave for 30 seconds and add active dry yeast and a teaspoon of sugar and give it a gentle whisk. Let it sit for about 5-10 minutes, until the mixture become foamy and bubbly.
In the bowl of a stand mixer with the dough hook attachment, add yeast mixture, flour, eggs, kosher salt, rosemary, and thyme. Start on low speed and gradually increase to medium speed for 3-4 minutes, until a cohesive dough forms and the dough pulls away from the sides of the bowl.
Increase the mixer speed to high and begin adding room temperature soft butter, one tablespoon at a time. Once all of the butter is added, continue mixing the dough for 8 minutes. Do the window pane test - take a piece of the dough and stretch it into a square until it's thin enough that you can see through it. It the dough breaks, continue kneading the dough for another 2-3 minutes. If it stays intact, roll the dough into a smooth ball and move onto the next step.
Spray a large bowl with oil. Add the dough to the greased bowl. Cover with plastic wrap and let the dough rise in a warm place for 1-1 ½ hours until the dough has doubled in size.
While the dough is rising, make the filling. Combine room temperature butter, thyme, sage rosemary, basil, and garlic in a bowl and mix together. Set aside.
Once the dough has risen, punch the dough to deflate it and place onto a lightly floured surface. Use a rolling pin to roll the dough into a 10x14 inch rectangle. Spread the butter filling all over the dough, then sprinkle the mozzarella and cheddar cheese on top.
Roll the dough tightly into a log and pinch the seam to seal in the filling. Cut off any excess dough on the ends.
Just below the top of the log, cut the log in half lengthwise with a sharp knife, leaving the top connected.
Wrap pieces of the log around each other and pinch the bottom to seal.Place in a 9x5 loaf pan and cover with a kitchen towel or plastic wrap. Allow it to rise again for about an hour, until it becomes puffy. (second rise)
Preheat the oven to 350 degrees F. Make the egg wash by mixing together egg yolk and milk. Brush the top of the babka with egg wash and bake for 35-40 minutes, until the tops are golden.