In a mixing 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.

Heat a non-stick skillet over medium-high heat, add the lean ground beef, cook and break it into pieces for about 6-8 minutes until browned. Drain excess grease, then season with salt, black pepper, garlic powder, and Worcestershire sauce.

On each tortilla, lay down 2 slices of cheddar cheese, followed by the ground beef, chopped onion, and pickles. Top with 2 more slices of cheese, fold the tortilla over.

Heat olive oil in the skillet, place folded quesadillas into the hot oil, cooking for 2-3 minutes on each side until golden-brown and cheese is melted.

Transfer quesadillas to a cutting board, let cool for a minute, slice into triangles and serve hot with the burger sauce.