To make the pickled red onions: In a medium bowl, combine all the ingredients. Set aside for 1 hour, stirring occasionally, until onions are pickled. (Keeps for 1 week in the refrigerator.)
To make the burger sauce: In a small bowl, combine all the ingredients. (Keeps for 1 week in the refrigerator.)
Divide the ground beef into 4 equal portions and pat them into 4-inch disks. Season both sides generously with salt and pepper. Brush cut sides of burger buns with the butter.
Grill the beef patties until both sides are nicely charred, turning occasionally, 5 to 6 minutes. About 1 minute before the burgers are done, place a slice of cheese on each one and grill the buns.
Generously spread the burger sauce on both the top and bottom halves of the buns. On each bottom bun, layer the lettuce, cheeseburger patty, a pile of pickled onions and the bread and butter pickles. Add ketchup to the top bun, if desired, and serve.
