In a small bowl, combine mayonnaise, grated onion, ketchup, pickle relish, yellow mustard, white vinegar, and paprika. Whisk until smooth. Cover and refrigerate to let the flavors meld.

Heat a large skillet over medium-high heat. Add the ground beef, breaking it apart as it cooks (about 5-7 minutes until browned and crumbly).

Drain any excess grease from the skillet. Return to heat, then mix in salt, black pepper, garlic powder, and Worcestershire sauce. Stir until everything is incorporated and fragrant.

Lay your tortillas flat. On one half of each tortilla, place 2 slices of cheese. Divide the seasoned beef evenly among the tortillas, sprinkle on the chopped onions and pickles, and then top with 2 more slices of cheese. Fold the empty half over to create a pocket.

Wipe out the skillet or grab a clean griddle. Heat the olive oil over medium-high. Carefully add the quesadillas, cooking for about 2-3 minutes per side until they turn golden brown and crispy, with the cheese oozing out.

Transfer to a cutting board and let rest for a minute. Cut into 2-3 triangular pieces and serve with the burger sauce for dipping.