In a small bowl, mix together the mayonnaise, ketchup, relish, and smoked paprika until well-blended. Cover the sauce and refrigerate it until you're ready to assemble the burgers.

Shape the patties: Divide the ground beef into four equal portions, about 4 oz each. Gently form each portion into a round ball, being careful not to overwork the meat.

Season and chill: Sprinkle salt and pepper over the meatballs, then place them on a plate lined with wax paper. Keep the meatballs cold in the fridge or freezer while you prepare the cooking pan.

Prepare the pan: Heat a cast iron skillet over medium-high heat. Add a teaspoon of oil and spread it evenly across the surface with a paper towel. Heat until the pan begins to smoke.

Smash the burgers: Place one meatball onto the hot pan. Immediately use a spatula or burger press to press down on the meatball, flattening it into a thin patty. If necessary, use a piece of parchment paper to prevent sticking.

Cook: Let the burger cook for about a minute until a crust forms. Resist the urge to press down again or flip too early. Season with more salt and pepper as it cooks.

Flip the burger: Carefully scrape underneath the patty to flip it, ensuring you keep the crust intact. If you're using cheese, add a slice now. Cook for another 45 seconds or until done. Transfer the burgers to a plate and cover with foil.

Toast the buns: Add butter to the pan and let it melt. Turn off the heat, then place the cut sides of the buns onto the pan to toast them lightly.

Assemble & enjoy: Spread the prepared sauce onto the toasted buns, add the burgers, and top with your favorite toppings. Serve immediately.