Prepare a gas grill with a grill pan for medium heat.
Gently form the ground beef into 4 loosely packed patties, making sure not to over work the meat. Season the burgers with salt and pepper.
Place 4 onion rings on the grill and press a burger patty onto each one. Let this cook together until the burgers caramelize on that side, 3 to 4 minutes. Flip the burgers and top with 2 slices cheese (on top of the onions), then continue to cook to your desired temperature.
Spread 1 tablespoon butter onto each English muffin. Season with salt and pepper, then toast on the grill.
In a mixing bowl, combine the Dijon, mayonnaise, ketchup, horseradish, gherkins and chives. Fold together until evenly incorporated.
To assemble, spread the special sauce on both slices of each English muffin. Stack the burgers on the bottom slices, then the pickles (if using), lettuce and top slices of English muffin.