In a small bowl, stir together mayonnaise, ketchup, mustard, chopped pickle, pickle juice, and smoked paprika until smooth. Taste and adjust salt if needed. Set aside so the flavors meld while you cook.
Place the ground beef in a bowl. Sprinkle with salt, pepper, and garlic powder. Gently mix just until combined—overmixing makes the patties tough. Divide into 8 equal portions (about 2 ounces each) and roll lightly into balls. Do not flatten yet.
Place a large cast-iron or heavy stainless-steel skillet over medium-high heat. Let it get very hot, about 2–3 minutes. A drop of water should sizzle and evaporate instantly.
Add a light film of oil if your skillet is not well-seasoned. Place 3–4 beef balls in the pan, leaving space between them. Immediately press each one firmly with a sturdy spatula or burger press until very thin (about ¼-inch thick). Cook undisturbed for 2–2½ minutes until the edges turn deep golden and crispy. Flip carefully, top each with a sprinkle of cheddar, and cook 1 more minute until the cheese melts and the second side browns. Transfer to a plate and repeat with remaining beef.
Divide the shredded lettuce among four bowls. Place two hot, cheesy smash patties on each bed of greens. Spoon a generous amount of special sauce over the top. Add red onion, pickle chips, diced tomato, and any optional toppings. Finish with a sprinkle of parsley if using. Serve immediately while the cheese is still stretchy and the edges stay crisp.