Prep the toppings: Rinse and dry the lettuce, slice the tomato and red onion, and have the dill pickle chips, ketchup, yellow mustard, and mayonnaise ready.
Toast the buns: Melt the butter in a large skillet over medium heat. Place buns cut side down and toast until golden, 1–2 minutes. Transfer buns to a tray. (If grilling, brush cut sides lightly with melted butter and toast on the grill 30–60 seconds.)
Form the patties: Divide the ground beef into 4 equal portions (about 6 ounces each). Gently form 3/4-inch-thick rounds slightly wider than the buns. Make a shallow 1-inch-wide dimple in the center of each patty; avoid overworking the meat.
Preheat the cooking surface: Heat the skillet to medium-high and add the vegetable oil (or oil the grill grates). Just before cooking, season both sides of the patties evenly with the kosher salt and black pepper.
Sear the burgers: Add patties to the hot skillet or grill. Cook without moving until a deep brown crust forms and juices begin to pool on top, 3–4 minutes. Flip and cook 2–4 minutes more to desired doneness—edges browned and centers springy; about 130–135°F for medium-rare, 135–145°F for medium, or 160°F for well-done.
Optional cheese: In the last minute of cooking, top each patty with an American cheese slice and cover the skillet or close the grill lid to melt.
Assemble and serve: Spread ketchup, yellow mustard, and mayonnaise on the toasted buns. Set a patty on each bottom bun and top with lettuce, tomato slices, red onion, and dill pickle chips. Cap with the top buns and serve immediately.