Gently heat the milk and butter in a small saucepan until the butter is melted. The milk should be just a little hotter than warm, between 100 and 115 degrees F (but no warmer than 115 degrees F). Remove the saucepan from the heat and whisk in the sugar, yeast and 2 teaspoons salt. Let the mixture sit until the yeast is activated and foam covers the top, 5 to 8 minutes.
Add the flour, egg and yeast liquid to a large bowl. Make the dough by mixing the ingredients together with a rubber spatula. Make sure all ingredients are incorporated; the dough will be sticky and loose. Cover 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 in order to deflate it while still in the bowl. Form into a ball, as best you can, and cover with plastic wrap. Let rest in a warm place until doubled in size, about 1 hour.
Line 2 baking sheets with parchment. Transfer 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 into a ball, incorporating more flour as needed so the dough is no longer sticky but still pliable. 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 the prepared baking sheets seam-side down. (It's ok if the filling is not fully encased at this point.) Repeat with the remaining pepperoni and cheese, placing the bundles on the baking sheets at least 1 inch apart to allow for a third rise. 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 the rolls with the remaining oil and bake until the crust is crispy and golden brown, 30 to 35 minutes.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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