Mix together the mayo, ketchup, and sweet relish to taste in a resealable container; Set aside.
Preheat a large skillet over medium-high heat. Cook ground beef, breaking into chunks, and season with salt, pepper and any other desired burger seasonings (I use McCormick Hamburger seasoning and Worcestershire sauce). Add the onions, cover, lower heat to medium, and cook 2-5 minutes longer until onions.
Wipe out skillet, or preheat quesadilla/panini press.
Place on tortilla on preheated quesadilla press. Layer lettuce, cheese, hamburger, onions over tortilla. Smear another tortilla with mayo mixture and top. Press lid closed and cook until cheese is melted and gooey.
Cut and serve.
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