Activate yeast by mixing warm milk, sugar, and yeast; wait 8-10 minutes until foamy.
Add egg, flour, and salt to the yeast mixture; mix until a dough forms.
Knead in softened butter, then let the dough rise in a greased bowl until doubled (~60-90 min).
Prepare garlic butter with butter, garlic, parsley, salt, and pepper. Roll out dough, spread garlic butter, add cheeses, then roll and cut into 8 pieces.
Arrange rolls in a greased pan, let rise (~25-35 min), then brush with egg wash and bake at 350°F (175°C) for 25-30 min until golden.
Melt remaining garlic butter and brush over warm rolls before serving.
Notes
For best results, ensure dough is slightly tacky and rolls are evenly spaced during baking.