In a large bowl, add flour, butter, yeast, sugar, salt, egg, milk and cheese, mixing with a spoon until dough is partially formed.
Using a stand mixer with the dough hook on, knead dough for 6 minutes. After kneading, it should feel soft to the touch. Dough shouldn't be too sticky.
Use a bit of oil to grease a clean bowl, so the dough will be easier to remove from it after the first rise. Place dough on it, cover with a damp towel or plastic wrap and let it rest in a warm spot for 1h30. It should double in size.
After first rise, deflate the dough, dividing it into 8 equal parts.
Fold the dough over itself 4 times, creating a ball. Pinch the ends together, placing them facing down.
Flatten each ball with the palm of your hand, covering them loosely. Let them rise for 30 to 45 minutes, or until buns are looking puffy and light.
Preheat the oven, 375ºF. Just before baking, whisk the egg yolk, brushing each bun with it before topping with the extra shredded cheese.
Bake for 15 to 17 minutes, or until the cheese has melted and the bun tops are golden.