How to Make It
Prepare the Hamburgers: Gently mix together ground beef and next 5 ingredients (through onion powder) in a bowl. Shape mixture into 3 (1/2-inch-thick) patties. Heat a large cast-iron skillet or grill pan over medium-high. Coat with cooking spray. Add patties to pan and cook to desired degree of doneness, about 2 to 3 minutes per side for medium. Remove patties from skillet; let stand until cool enough to handle.
Prepare the Egg Rolls: Break Hamburger patties into 1/2-inch chunks. Toss together Hamburger chunks, lettuce, pickles, mayonnaise, and mustard in a bowl until just combined.
Whisk together egg and water in a small bowl. Place 4 of the egg roll wrappers on a work surface with the point of 1 corner of each facing toward you. Place 1/2 cup Hamburger mixture in center of each wrapper; shape into a 4x1-inch horizontal log. Top each with 1 cheese slice half. Fold bottom point of each wrapper over filling. Fold 2 sides over filling. Brush remaining top point with a small amount of the egg wash. Roll up egg roll from bottom to seal. Lightly dust a tray with cornstarch. Place filled egg rolls on tray. Repeat process with remaning egg roll wrappers, Hamburger filling, cheese slice halves, and egg wash. Cover with plastic wrap, and chill until ready to fry.
When ready to fry egg rolls, heat oil in a large stockpot or Dutch oven over high to 360°F. Add egg rolls; cook until golden brown on all sides, 3 to 4 minutes. Transfer to a plate lined with paper towels to drain. Serve with Sweet and Sour Ketchup.
Prepare the Sweet and Sour Ketchup: Stir together cornstarch and 2 tablespoons of the water in a small bowl until cornstarch dissolves. Whisk together cornstarch mixture, ketchup, vinegar, sugar, soy sauce, salt, and remaining 1 cup water in a saucepan. Bring to a boil over medium-high. Reduce heat to medium, and simmer, stirring occasionally, until slightly thickened, 2 to 3 minutes.