Prepare the Filling
Brown the ground beef in a large skillet over medium heat until no pink remains, then drain excess fat. Add the diced onions and continue cooking for 2 minutes until slightly softened. Remove from heat and stir in the shredded cheese, diced pickles, salt, and pepper until the cheese begins to melt. Set aside and allow the mixture to cool completely before assembling the egg rolls.
Make the Egg Wash and Sauce
In a small bowl, whisk together the egg and water to create your egg wash for sealing the wrappers. For the Big Mac sauce, combine mayonnaise, ketchup, mustard, sweet pickle relish, minced onions, paprika, vinegar, onion powder, and salt in a separate bowl. Whisk until all ingredients are thoroughly combined into a smooth sauce.
Assemble the Egg Rolls
Lay an egg roll wrapper on a clean work surface in a diamond orientation. Lightly brush the edges with egg wash. Place 3-4 tablespoons of the cooled beef filling on the lower third of the wrapper. Fold the bottom corner over the filling, then tuck it under and press down to compact the filling. Fold the right and left corners toward the center, creating an envelope shape.
Seal the Egg Rolls
Brush the top corner with egg wash, then roll the egg roll upward tightly until completely sealed. Make sure all edges are secure to prevent filling from leaking during frying. Repeat this process with all remaining wrappers and filling.
Fry the Egg Rolls
Heat oil to 350°F in a large, deep skillet or pot. Working in batches of 5-6 egg rolls at a time to avoid overcrowding, carefully lower them into the hot oil. Fry for 3-4 minutes per side until they turn golden brown and crispy all over. Maintain oil temperature between batches for consistent results.
Drain and Serve
Remove fried egg rolls from the oil using a slotted spoon or tongs and place them on paper towels to absorb excess oil. Then transfer to a wire rack to prevent them from becoming soggy on the bottom. Serve hot with the prepared Big Mac sauce for dipping.