Caramelized Onions: Heat oil in large nonstick skillet set over medium-high heat; cook onion, sugar, vinegar, salt and pepper, stirring occasionally, for about 20 minutes or until soft and caramelized. Let cool.
Dijonnaise Sauce: Whisk mayonnaise with mustard. Set aside.
Pork Burgers: Mix together pork, egg, bread crumbs, Worcestershire, Dijon, thyme, salt and pepper; form mixture into 8 patties. Divide onions and cheese among 4 patties; top with remaining patties, sealing edges to fully enclose filling. Flatten patties to 1/2-inch thickness.
Preheat your Kenyon Grill to medium-high. Place burgers on grill and close lid. Cook, turning once, for 15 minutes or until meat is no longer pink inside. Serve on kaiser rolls with Dijonnaise Sauce and arugula.