In a small saucepan, gently heat the milk and butter until the butter is melted. The milk should be just a little hotter than warm, between 100 and 115 degrees F, but not over 115 degrees F. Remove the saucepan from the heat and whisk in the sugar, salt and yeast. Let the mixture sit until the yeast is activated and foam covers the top, 5 to 8 minutes.
Add the flour to a large bowl and make a well in the center. Crack the egg into the middle and pour in the yeast liquid. Make the dough by mixing all ingredients together with a rubber spatula. Make sure all ingredients are incorporated; the dough will be sticky and loose. Leave the dough in the center of the bowl when it is fully incorporated. Cover the bowl with plastic wrap and let rise in a warm place until doubled in size, about 2 hours. Remove the plastic and gently re-knead the dough while still in the bowl. Form into a ball, as best you can, and cover with plastic wrap. This time allow to rest in a warm place for 1 hour.
After the second rise, remove the dough to a very generously floured surface, kneading to bring together. Cut the dough into 8 pieces, about 3 1/2 ounces each. Gently form each piece of dough into balls, incorporating more flour as needed. Use your hands to flatten each ball to a 4 1/2-inch circle. Brush a piece of pepperoni with oil and place in the center of the circle, along with 2 tablespoons shredded mozzarella. Fold the dough over the pepperoni, like a burrito, and place on a parchment-covered baking sheet seam side down. Repeat with the remaining pepperoni and cheese. As you place each pepperoni roll on the sheet tray, leave at least 1 inch or space around each roll to allow for a third rise (therefore you will need 2 baking sheets). Cover the rolls loosely with plastic wrap and place in a warm place for 30 minutes. The rolls will puff up just a bit.
Preheat the oven to 350 degrees F. Brush the tops of rolls with the remaining oil and bake in the oven 30 to 35 minutes. The rolls will have a rich golden color and crispy crust.
This dough is very wet, and will look too loose when you first make it. Use a rubber spatula when incorporating flour into the dough. Make sure to always flour your hands and the surface you are working on when working with the dough.