Preheat oven to 375°F (190°C) and lightly grease a baking dish
In a bowl, combine ground beef, egg, breadcrumbs, garlic, onion, paprika, salt, and pepper
Mix gently until just combined without overworking the meat
Shape mixture into evenly sized patties
Heat olive oil in a skillet over medium heat
Sear patties on both sides until lightly browned
Transfer patties into the prepared baking dish
Melt butter in the same skillet and whisk in flour
Gradually add milk and beef broth while whisking
Cook until the sauce thickens and becomes smooth
Stir in mozzarella and cheddar until melted
Pour sauce around the patties in the baking dish
Top patties generously with mozzarella cheese
Bake for 20–25 minutes until cheese is bubbly and golden
Garnish with fresh parsley before serving