In a small bowl, combine mayonnaise, grated onion, ketchup, pickle relish, yellow mustard, white vinegar, and paprika. Whisk until smooth and well blended. Cover and refrigerate until ready to use.
Heat a large skillet over medium-high heat. Add the ground beef and cook, breaking it apart into small crumbles. Continue cooking until the beef is completely browned, about 5-7 minutes.
Drain any excess grease from the skillet. Return the skillet to the heat and add the salt, black pepper, garlic powder, and Worcestershire sauce to the beef. Stir well to evenly distribute the seasonings. Remove from heat once fully incorporated.
Lay the tortillas flat on your work surface. On one half of each tortilla, place 2 slices of cheese. Divide the seasoned beef mixture evenly among the tortillas, placing it on top of the cheese. Sprinkle each with equal amounts of chopped onions and pickles. Top each with 2 more slices of cheese. Fold the empty half of each tortilla over the filling.
Wipe out the skillet or use a clean griddle. Heat over medium-high heat and add the olive oil. When the oil is hot, carefully place the quesadillas in the pan. Cook for 2-3 minutes per side until golden brown and crispy, and the cheese is melted.
Transfer the quesadillas to a cutting board and let rest for 1 minute. Cut each quesadilla into 2-3 triangular pieces. Serve immediately with the prepared burger sauce on the side for dipping.
