Mix all burger sauce ingredients together until combined. Cover and let sit in the fridge while you make the rest of the ingredients. The longer the sauce sits, the deeper the flavor gets.
Set a medium skillet over medium heat. Add oil. Once hot, add ground beef. Break down beef until it's in small pieces. Cook until brown. Drain out excess oil. Return to heat and add Worcestershire, mustard, and a little salt and pepper. Cook for a minute or so. Remove from heat.
Place a cast-iron skillet or Dutch oven on the stove. Add about 2 inches of oil and set over medium heat. While the oil heats up, make the egg rolls.
Grab one wonton wrapper and place on a clean counter in front of you, with a corner pointing towards you. Place a quarter cup of ground beef in the center of the wonton. Place half a slice of American cheese on top of the beef, along with two slices of pickle.
Fold over the bottom corner over the center and gently, yet tightly pull over the center. Fold over the right and left corners of the wonton wrapper over the middle so they meet in the center. Brush a little water on the edges to seal them. Brush water on the top corner and roll it around the egg roll to seal it closed. Repeat with other 9 wontons.
Once oil is 350 degrees, add an egg roll and fry about 1 minute per side or until deep golden brown. Remove hot oil and drain on a rack until warm to the touch. Serve with burger sauce.