In a small bowl, whisk together the mayonnaise, yellow mustard, sweet pickle relish, smoked paprika, and garlic powder until smooth. Set the sauce aside in the refrigerator to let the flavors marry.

Spread softened butter onto the cut sides of the potato buns. Heat a large cast-iron skillet or griddle over medium heat and toast the buns cut-side down until golden brown and warm, about 1 to 2 minutes. Transfer to plates and spread a spoonful of burger sauce on the bottom buns.

Turn the heat up to high. Let the cast-iron skillet heat up until it is lightly smoking. Do not add any oil to the pan; a dry, hot skillet is crucial for getting that signature lacy crust.

Place 2 to 4 cold beef balls into the skillet, leaving ample space between them. Immediately press each ball down with your metal spatula, using your second tool to apply firm, even pressure until the patty is extremely thin (less than 1/4 inch) and flat. Season generously with salt and pepper.

Cook undisturbed for about 2 minutes, until the edges are deeply browned, crispy, and starting to lift. Slide a sharp metal spatula completely under the patty—scraping hard to keep the seared crust attached to the meat—and flip it over.

Immediately place a slice of American cheese onto each patty. Cook for an additional 45 to 60 seconds until the cheese melts and the patty is cooked through. Stack one cheesy patty on top of another.

Transfer the double patty stack onto a dressed bottom bun. Top with shredded lettuce and dill pickle chips, then cap with the top bun. Repeat the cooking process for the remaining patties and serve immediately.