Microwave 1 cup milk in 30-second increments until it reaches 100℉ to 110℉, or warm in a small saucepan on the stove over medium-low heat.
In a large mixing bowl or stand mixer fitted with a dough hook, combine the warm milk, 1 tablespoon instant dry yeast, 2 tablespoons granulated sugar, 1 teaspoon salt, 3 tablespoons salted butter, 1 large egg, and 2 1/2 cups of the flour.
Mix on low speed, then increase to medium once flour starts to incorporate. Slowly add more flour, a little at a time, until the dough pulls away from the sides of the bowl. You'll use between 3 to 3½ cups total. The dough should be soft, elastic, and tacky but not sticky.
Transfer the dough to a large lightly greased bowl. Cover with a towel or plastic wrap and set aside to rise until double in size, about 90 minutes.
Once the dough has doubled in size, in a small microwave safe bowl, melt 1/2 cup salted butter. Set aside to cool while you shape your rolls.
Lightly grease a 9x13 pan or 12-inch cast iron skillet with nonstick cooking spray. Divide the dough into about 12 equal pieces and shape into rolls. Shape each dough piece into a ball by rolling it on the counter. Use your thumbs to gently press open the center. Add a pinch of shredded mozzarella, then pinch the dough closed to seal.
Stir 4 teaspoons minced garlic and 1/2 cup grated parmesan cheese into the melted butter. Dip each roll into the garlic butter to coat and place into the prepared pan. Tuck extra shredded mozzarella between each roll for that cheesy pull-apart effect. Spoon any remaining garlic butter over the tops.
Cover again and set aside to rise until doubled in size, about 1 hour.
Preheat the oven to 375°F (190°C). Bake the rolls for 18 to 20 minutes or until lightly browned.
Serve warm or allow to cool completely. Once cooled completely, store in a plastic bag for 3 to 4 days.
Video
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Notes
No mixer? No problem! Start out by stirring with a spoon until it becomes too difficult to stir. Turn the dough out onto a clean surface and knead it with your hands, gradually adding in more flour and folding, pressing, and working the dough (kneading) until you have a soft and smooth dough ball that is tacky, but not sticky enough to stick to your hands.
To shape rolls, place your hand into a cupping shape parallel to a clean countertop. Place the dough ball between the palm of your hand and the countertop. Roll the dough in a circular motion between your palm and the countertop for approximately 15 to 30 seconds until you have a smooth, round ball.