In a small bowl, lightly beat 2 of the eggs with a fork and set aside.
In the bowl attachment of a stand mixer combine the milk and yeast (Make sure the milk is between 100° and 105° -- too hot and it will kill the yeast - and the rolls won't rise). Use a conventional whisk to whisk together the yeast and the milk. Add the sugar, salt, beaten eggs, butter, and 1 cup of the bread flour and the parmesan cheese. Whisk together to combine.
Add the remaining 3 1/2 cups of bread flour and place the mixing bowl in the stand mixer. Attach the dough hook and knead the flour into the yeast mixture on medium speed for 6-7 minutes or until dough is soft and elastic.
Lightly oil a large mixing bowl with canola oil and set the dough in the center, turning once or twice to get a light sheen of oil on the dough. Cover the dough with a tea towel and set it in a cold oven to proof for one hour, or until doubled in size.
Spray vegetable spray in a large rectangular pan (I used my lasagna pan 11" x 15") or use two smaller pans. Divide the dough into quarters. Divide each quarter into sixths and roll each into a ball. Place in the pan so that the dough balls do not touch.
Lay a piece of wax paper over the rolls and set in a cold oven to rise for another hour. When the dough has doubled in volume, remove it from the oven and preheat the oven to 375°.
Lightly beat the remaining egg and brush it over the rolls. The egg wash gives the rolls a sheen. When the oven is hot, bake the rolls for 20 minutes.
Remove the rolls from the oven and brush with melted butter. Sprinkle with parmesan cheese and parsley. Serve.