In a medium bowl, combine mayo, grated onion, ketchup, pickle relish, yellow mustard, white vinegar, and paprika. Whisk together until smooth and refrigerate for at least 15 minutes.

In a large skillet over medium-high heat, add lean ground beef and cook for 5-7 minutes until browned. Drain excess grease and season with salt, black pepper, garlic powder, and Worcestershire sauce. Stir and set aside.

Lay out two tortillas. On one half of each, layer cheddar cheese, the beef mixture, chopped onions, and pickles if using. Top with more cheese and fold the tortillas.

In the same skillet, heat olive oil over medium-high heat. Cook each folded quesadilla for 2-3 minutes on each side until golden brown and cheese is melted. Remove and repeat with remaining quesadillas.

Slice the cooked quesadillas into wedges and serve warm with the burger sauce.