MAKE THE BIG MAC SAUCE: In a small bowl, stir together 1/2 cup mayonnaise, 2 tablespoons ketchup, 1 tablespoon yellow mustard, 1 tablespoon sweet pickle relish, 1/2 teaspoon smoked paprika, 1/4 teaspoon kosher salt, and 1/8 teaspoon fresh-ground black pepper until smooth and one even color. Taste it and add a pinch more salt or pepper if you like. You will have about 3/4 cup. Set it aside.
COOK THE BEEF AND ONION: Put 1 pound 85% lean ground beef and half of the 1 small white onion, diced and divided (about 1/4 cup) in a large skillet over medium heat. Break the beef into small crumbles with a spoon as it cooks, until there is no pink left anywhere and the onion is soft, about 8 to 10 minutes. Tip the pan and spoon off the fat that collects on the low side. Draining keeps the sauce from turning greasy.
SEASON AND SIMMER: Stir in 1/2 cup ketchup, 1 tablespoon Worcestershire sauce, 1 teaspoon garlic powder, 1 teaspoon smoked paprika, 1/2 teaspoon kosher salt, and 1/4 teaspoon fresh-ground black pepper until the beef is evenly coated. Let it simmer until the sauce thickens and clings to the beef instead of pooling in the pan, about 2 to 3 minutes.
MAKE IT CREAMY: Turn the heat to medium-low. Stir in half of the Big Mac Sauce (about 6 tablespoons) and 4 ounces sharp cheddar cheese, shredded until the cheese melts completely and the beef looks creamy, about 1 to 2 minutes. Take the skillet off the heat.
TOAST THE BUNS: Set the 4 sesame seed hamburger buns cut side down in a dry skillet over medium heat. Toast until the cut sides are golden brown and firm, about 1 to 2 minutes. A toasted bun stays sturdy under the saucy beef.
BUILD THE SANDWICHES: Spread about 1 tablespoon of the remaining Big Mac Sauce on each bottom bun. Layer on 1/4 cup shredded iceberg lettuce and 1 slice American cheese, then spoon a quarter of the warm beef over the cheese. Top each with some of the remaining diced white onion (about 1 tablespoon each), 2 tablespoons dill pickle chips, and a drizzle of the last of the sauce. Add the top buns and serve while the beef is hot.
Notes
STORAGE: Store the leftover beef mixture in an airtight container in the refrigerator, with the sauce, toppings, and buns kept separate so nothing gets soggy. The beef is at its best for 3 days and safe to eat for up to 4 days.REHEATING: Reheat the beef mixture in a microwave-safe bowl in 30 second bursts, stirring between each one, until hot all the way through, about 1 to 2 minutes. Toast fresh buns and build the sandwiches just before serving.INGREDIENT TIP: 90% lean ground turkey or ground chicken works in place of the beef. Cook it the same way until there is no pink left and it reaches 165 degrees F on a meat thermometer.