Blend shredded cheddar, mozzarella, and softened cream cheese in a bowl until smooth, then shape into 2 to 3 tablespoon balls and freeze until firm.
Combine ground beef with garlic powder, onion powder, paprika, salt, and pepper in a large bowl without overmixing.
Divide the beef mixture into six equal portions and flatten each into thin patties.
Place a frozen cheese ball in the center of one patty, cover with another patty, then seal edges firmly to form a cake.
Preheat oven to 375°F (190°C). Melt butter in an oven-safe skillet over medium-high heat.
Sear each beef cake for 2 to 3 minutes per side until golden-brown crust forms.
Transfer skillet to the oven and bake for 10 to 12 minutes.
Allow beef cakes to rest for several minutes before slicing.
Optionally brush tops with BBQ sauce or ketchup glaze and serve immediately while cheese center is molten.