In a small bowl, combine 1/4 cup mayonnaise, 2 tablespoons ketchup, 1 teaspoon yellow mustard, 1 tablespoon sweet pickle relish, 1 teaspoon apple cider vinegar, 1/2 teaspoon Worcestershire sauce, 1/4 teaspoon garlic powder, 1/4 teaspoon paprika, 1/8 teaspoon salt, and a pinch (about 1/20 tsp) freshly ground black pepper. Whisk until smooth and homogeneous. Taste and adjust seasoning (more relish for sweetness, more vinegar for tang). Chill in refrigerator while preparing the rest to let flavors meld.

Veg & Pickles

Prepare vegetables

Thinly slice about 1/2 cup sweet onion into rings or half-rings. Rinse and dry romaine leaves and stack leaves so you have approximately 1 cup loosely packed (two leaves per burger). Drain dill pickle slices and pat dry if very wet. Keep all refrigerated until assembly.

Burger

Season patties

Pat the two 4-oz beef patties lightly with your hands to flatten slightly to about 3/4" thickness. Season both sides of each patty evenly with 1 teaspoon kosher salt and 1/2 teaspoon freshly ground black pepper total (divide evenly between patties). Let sit at room temperature for 5 minutes while you preheat your cooking surface.

Cook patties

Preheat a cast-iron skillet, griddle, or heavy pan over medium-high heat until shimmering (about 3–5 minutes). Lightly oil the pan if not well-seasoned. Place patties in the hot pan leaving space between them. Cook undisturbed for 3–4 minutes to develop a crust. Flip once and immediately place a slice of American cheese on each patty. Cook another 2–3 minutes for medium (internal temp ~160°F/71°C). If you prefer, cover the pan briefly (30–60 seconds) after adding cheese to help it melt. Remove patties and rest for 1–2 minutes.

Toast buns

While patties rest or in a separate dry pan over medium heat, spread softened butter (about 1/2 tablespoon divided) on the cut sides of the buns. Place buns cut-side down in the skillet or on a griddle for about 30–60 seconds until golden and slightly crisp. Remove from heat.

Assemble burgers

Spread about 1 to 1 1/2 tablespoons of the prepared house sauce on the bottom half of each toasted bun. Place a crisp romaine leaf (or two small leaves) on top of the sauce, then add the cheesy beef patty. Top each patty with 3 dill pickle slices and a generous portion (about 2 tablespoons) of thinly sliced sweet onion. Spread a small amount of sauce (about 1/2 tablespoon) on the top bun and close the burger. Press lightly to compact.

Serve

Serve immediately while hot. Optionally accompany with fries or chips. Any leftover sauce can be refrigerated in an airtight container for up to 5 days.