In a large skillet over medium-high, cook the ground beef until completely browned.
Drain the beef and then return it to the skillet. Add in the red onion, ketchup, mustard, dill pickled relish, and Worcestershire sauce. Stir the mixture together until evenly combined. Salt and pepper the mixture, to taste, and let it cook just until heated through.
Lay the tortillas out flat. Sprinkle some of the shredded cheese evenly out over one half of each tortilla. Spread 1/4 of the seasoned ground beef overtop of the cheese. Sprinkle another layer of cheese. Fold over the other half of the tortilla to close.
Spray a large skillet with non-stick spray and heat over medium heat.
Carefully transfer one of the filled until golden brown and crispy on one side, then flip and cook until golden brown on the other side.
Transfer the cooked quesadilla to a cutting board to cool, and repeat the previous steps until all of them are cooked and cooled.
Use a pizza cutter or sharp knife to cut them int wedges before serving.
Serve with ketchup and mustard for dipping, and with any other burger toppings you like.