01 - Combine warm milk, sugar, and yeast in a large bowl. Let stand for 5-10 minutes until mixture becomes foamy and fragrant.
02 - Whisk melted butter, egg, and salt into the yeast mixture. Gradually add flour, mixing until a soft, slightly sticky dough forms.
03 - Turn dough onto a lightly floured surface. Knead for 7-8 minutes until smooth, elastic, and no longer sticky.
04 - Place dough in a greased bowl, turning to coat all sides. Cover with plastic wrap and let rise in a warm, draft-free area for 1 hour until doubled in size.
05 - While dough rises, combine softened butter, minced garlic, mozzarella, Parmesan, parsley, and salt in a medium bowl until well blended.
06 - Punch down risen dough to release air. On a floured surface, roll into a 12x16-inch rectangle.
07 - Spread garlic cheese mixture evenly over the entire surface of the dough, leaving a 1/2-inch border at the edges.
08 - Starting from one long edge, roll dough tightly into a log, sealing the seam with your fingers.
09 - Cut log into 12 equal pieces. Place rolls cut-side up in a greased 9x13-inch baking dish, spacing evenly.
10 - Cover dish with a towel and let rise for 25-30 minutes until puffy and rolls are touching. Preheat oven to 350°F during final 10 minutes of rising.
11 - Bake for 22-25 minutes until rolls are golden brown on top and cooked through.
12 - Immediately brush hot rolls with melted butter. Sprinkle with fresh parsley and Parmesan cheese. Serve warm.