01 - Peel onions and slice into thick 1/2-inch rings. Separate into individual rings and set aside.
02 - Place flour in one shallow bowl. Beat eggs in a second bowl. Combine panko breadcrumbs, smoked paprika, garlic powder, salt, and pepper in a third bowl.
03 - Heat olive oil in skillet over medium heat. Add ground beef, salt, pepper, and Worcestershire sauce. Cook until browned, approximately 5-7 minutes. Remove from heat and cool slightly.
04 - Fold pickles, tomatoes, shredded cheddar, ketchup, and mustard into the cooled beef mixture until well combined.
05 - Select the largest intact onion rings. Fill each ring with cheeseburger mixture, pressing gently to create compact filling within each ring.
06 - Dredge each filled ring in flour, shaking off excess. Dip into beaten eggs, then coat thoroughly with seasoned panko mixture, pressing gently to adhere.
07 - Heat vegetable oil in deep fryer or heavy pot to 350°F. Fry stuffed rings in batches for 2-3 minutes per side until golden brown and crispy. Drain on paper towels.
08 - Arrange hot rolls on serving platter. Accompany with burger sauce or dipping sauce of choice.