In the bowl of a stand up mixer mix add the flour, salt, sugar, olive oil, 3 tablespoons parmigiano cheese (20 grams) and yeast then add a ¼ cup / 60 grams of water, start to mix with a fork.
Add the beaten egg, remaining parmigiano 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.
Notes
How to store the cheese buns
Place any leftover cheese buns in an airtight container or bag and keep at room temperature for up to 2-3 days. For longer storage, freeze the completely cooled buns in a freezer bag or container for up to two months. To reheat, loosely wrap them in foil and warm in a 350F oven.