Combine milk, water, and butter in a small saucepan. Heat over medium heat until liquids are very warm. Remove from heat.

Place 3 cups of flour, sugar, salt, and yeast in the bowl of a stand mixer fitted with the hook attachment. Mix just until dry ingredients are combined. Gradually add whole egg, egg yolk (reserve the white), and liquids. Mix on medium-low speed just until ingredients are combined, then add in the remaining flour, 1/4 cup at a time, until dough clings to the hook and pulls away from the sides of the bowl, about 2 minutes. Continue kneading on medium speed for an additional 2 minutes.

Place dough in a greased bowl, turning to lightly grease the top. Cover with a clean towel and let rise until doubled in bulk, about 1 hour.

While dough rises, make the garlic butter: Mix melted butter, garlic, and dried herbs in a small bowl. Let sit in a cool spot while dough rises so that butter starts to solidify again, making it easier to spread later.

Once dough has risen, turn out onto a lightly-floured surface. Roll into a 9x14-inch rectangle. Spread with the garlic butter, then sprinkle with the cheese. Starting from the short end, roll the dough tightly into a loaf and place in a greased 9x5-inch loaf pan. Cover with the towel and allow to rise until doubled again, about 30-40 minutes.

Near the end of the second rise, preheat oven to 375 degrees. Bake loaf for 40 minutes. Brush with the reserved egg white, then bake for an additional 5 minutes, or until deeply browned. Remove from pan to a cooling rack immediately. Allow to cool slightly before cutting and serving.