Combine beef, onion, garlic, soy sauce, salt, and pepper in a bowl, form mixture into patties. Heat oil in a large skillet over medium-high; add patties, and cook until cooked through, about 6 minutes per side. Remove from pan and let cool.
Break burgers into chunks. Gently toss together burger pieces, pickles, mayonnaise, and mustard in a bowl until combined. Place egg roll wrappers on a work surface with a point towards you. Have a small bowl of water near by for assembly. Place 1/8 cup burger mixture in the middle of 1 wrapper; top mixture with a sprinkle of cheese. Fold point closest to you over mixture, tucking in tightly. Brush all exposed sides of wrappers with water. Fold over sides, pressing to adhere; roll over to seal. Place on a baking sheet dusted with cornstarch. Repeat procedure with remaining burger mixture and wrappers. Cover with plastic wrap and chill until ready to use.
Prepare the Special Sauce
Combine ketchup, mayonnaise, soy sauce, vinegar, sugar, and cayenne pepper in a bowl; cover and chill until ready to use.
Cooking the Egg Rolls
Heat vegetable oil in a nonstick pan over medium-low heat. Add egg rolls; cook, turning occasionally, until golden brown, about 2 minutes. Drain on paper towels and sprinkle with sesame seeds. Serve with special sauce.