Activate yeast in warm milk with sugar until foamy.
Mix flour, salt, and butter into yeast mixture; knead until smooth.
Let dough rise until doubled.
Brown ground beef; season and drain; cool.
Divide dough; roll into circles.
Add beef and cheeses to center of each circle.
Pinch edges tightly to seal into balls.
Place seam-side down in pan; brush with beaten egg.
Bake until golden brown.
Sprinkle with parsley and serve warm with sauce.
Notes
Store leftovers in an airtight container up to 3 days. Reheat in oven or microwave to restore meltiness. Freeze unbaked sealed buns for later baking. Adjust cheese amounts for stronger melt or milder flavor.