In the bowl of a stand up mixer mix add the flour, salt, sugar, olive oil, 3 tablespoons Parmesan cheese (20 grams) and yeast then add a 1/4 cup of water (58 grams) then start to mix with a fork.
Then add the beaten egg, remaining Parmesan cheese and Provolone and remaining water. Combine with a fork just until the dough starts to come together. Then add the dough hook attachment and knead for approximately 5 minutes or until smooth and elastic. (see below for hand kneading instructions). Remove dough to a lightly floured flat surface.
Form the dough into a ball, place in lightly oiled medium-sized bowl (make a couple of slashes on top with a knife). Cover well (I use a folded tablecloth) and let rise in a draft free area for 1-2 hours or doubled in bulk. (mine actually sat for 3 hours).
Remove the dough from bowl and on a lightly floured flat surface knead dough a couple of minutes, shape into 6 bun shapes, score with a knife and place on a lightly oiled parchment paper lined cookie sheet, cover and let rise for 1 hour.
Pre-heat oven to 400° (200° celsius).
Brush buns with milk and sprinkle with additional Parmesan cheese or grated Provolone. Bake for approximately 30 minutes. Let cool or eat warm. Enjoy!
KNEADING BY HAND
Place 1 tablespoon of flour on flat surface, place the dough, start to knead, add 2 more tablespoons of flour, one at a time while continuing to knead. Knead until smooth and elastic.