In a small bowl, whisk together the mayonnaise, ketchup, diced pickles, white vinegar, garlic powder, onion powder, and paprika until well combined. Set aside.

Divide the ground beef into four equal portions (about ¼ lb each) and gently roll them into balls.

Heat a large cast-iron skillet over medium-high heat. Place a beef ball on the hot surface and immediately use a sturdy metal spatula to smash it into a very thin patty. Season generously with salt and pepper. Cook for 2-3 minutes until a brown crust forms.

Flip the patty. Immediately place one tortilla on top, followed by ½ cup of shredded cheese. Cook for 1-2 minutes to let the cheese start melting.

Using a spatula, flip the entire assembly so the tortilla is on the bottom. Spread special sauce over the beef, then fold the tortilla in half. Add butter to the pan if needed and cook for 2-3 minutes per side until golden brown and crispy.

Remove from the skillet, let rest for a minute, then slice. Repeat for the remaining quesadillas and serve hot with extra sauce.