In a small bowl, whisk together mayonnaise, ketchup, yellow mustard, sweet pickle relish, white vinegar, and smoked paprika until smooth. Set aside.
Heat a large skillet over medium-high heat. Add the ground beef and diced onion. Cook, breaking up the meat with a spatula, until the beef is browned and the onion is soft, about 6-8 minutes. Drain any excess fat. Season with garlic powder, onion powder, paprika, salt, and pepper. Remove from heat and let cool slightly.
Lay out four tortillas on a clean surface. On each, sprinkle 1/4 cup of shredded cheese. Divide the cooked beef mixture evenly over the cheese. Top with chopped pickles and drizzle with 1-2 tablespoons of the burger sauce. Sprinkle another 1/4 cup of cheese over each, then place a second tortilla on top to form a sandwich.
Wipe the skillet clean and heat 1/2 tablespoon of oil over medium heat. Carefully place one assembled quesadilla in the skillet. Cook for 2-3 minutes per side, pressing down gently with a spatula, until the tortilla is golden brown and crispy and the cheese is melted. Repeat with remaining quesadillas, adding more oil as needed.
Transfer each cooked quesadilla to a cutting board and let rest for 1 minute. Slice into quarters or six wedges. Serve immediately, garnished with shredded lettuce and diced tomato, with extra burger sauce on the side for dipping.