01 - Combine warm milk, sugar, and yeast in a large bowl. Let sit for 5–10 minutes until foamy and bubbly.
02 - Add melted butter, egg, and salt to the yeast mixture. Gradually incorporate flour until a soft, sticky dough forms.
03 - Turn dough onto a floured surface and knead for 7–8 minutes until smooth, elastic, and no longer sticky.
04 - Place dough in a greased bowl, cover with plastic wrap or towel, and let rise in a warm place until doubled in volume, about 1 hour.
05 - While dough rises, mix softened butter, minced garlic, mozzarella, Parmesan, parsley, and salt until well combined.
06 - Punch down risen dough and roll into a 12x16-inch rectangle. Spread cheese filling evenly over the surface.
07 - Starting from a long edge, roll dough tightly into a log, sealing the seam.
08 - Cut log into 12 equal pieces and place cut-side up in a greased 9x13-inch baking dish.
09 - Cover dish and let rolls rise for 25–30 minutes until puffy. Preheat oven to 350°F during the final 10 minutes.
10 - Bake for 22–25 minutes until golden brown and cooked through.
11 - Brush hot rolls immediately with melted butter, then sprinkle with parsley and Parmesan. Serve warm.