01 - Heat a large skillet over medium heat. Add ground beef and season with salt and pepper. Cook until thoroughly browned, breaking it apart with a spoon for approximately 5 to 7 minutes. Incorporate the diced onion and sauté for an additional 2 to 3 minutes until softened. Stir in the Worcestershire sauce and chopped pickles. Remove from heat and allow to cool slightly.
02 - Gently mix the shredded cheddar cheese into the cooled beef mixture until evenly distributed.
03 - Place an egg roll wrapper on a clean, dry surface with a corner oriented towards you. Position approximately 2 to 3 tablespoons of the beef filling mixture near the center. Fold the bottom corner upwards to cover the filling. Fold in the left and right sides. Tightly roll the wrapper upwards, using the beaten egg to seal the final edge.
04 - Heat 2 inches of vegetable oil in a deep skillet or pot to a temperature of 350°F (175°C). Carefully fry the assembled egg rolls in batches for 2 to 3 minutes per side, ensuring they achieve a golden-brown and crispy texture. Remove from the hot oil and drain on paper towels.
05 - In a small bowl, vigorously whisk together the ketchup, mayonnaise, yellow mustard, pickle juice, and smoked paprika until a smooth consistency is achieved.
06 - Present the cheeseburger egg rolls warm, accompanied by the prepared dipping sauce.