These are as voluptuous and sexy as burgers get. They have a personalized ground meat mix with three kinds of beef, plus bacon and extra beef fat. You don't need a meat grinder, as the partially frozen meat is chopped in a food processor. Searing the burgers in a smoking hot pan gives them that flavorful crust that you just don't get from grilling over coals. And a specially made, "fancy" sauce like this one is never a bad idea.
To make the patties: Cut the short ribs, ribeye, chuck, the fat from the ribeye, and bacon into 1-inch chunks. Toss together on a baking sheet and freeze until partially frozen, 1 to 2 hours. In three or four batches, pulse all the meat and fat in a food processor until the mixture is finely ground but not a paste. Transfer the ground meats to a medium bowl, season with the salt and pepper, and mix gently. Divide the mixture into 12 equal portions and shape into patties about 2 inches wide. Alternatively, you can use a ring mold to shape the patties. (The patties can be covered and refrigerated for up to 6 hours. Remove from the refrigerator 30 minutes before cooking.)
To make the sauce: Mix all of the ingredients together in a medium bowl until combined. (The sauce can be covered and refrigerated for up to 3 days.)
To braise the onions: Melt the butter in a large skillet over medium heat. Add the onion and sugar and cook, stirring occasionally, until the onion is lightly browned, 8 to 10 minutes. Add the cola, bring to a boil, and cook until the liquid has reduced to a glaze, about 5 minutes. Season to taste with salt and cover the skillet with its lid to keep the onions warm. (The onions can be cooled, covered, and refrigerated for up to 1 day. Reheat the onions in the skillet before using.)
To cook the burgers: Lightly season the patties all over with salt and pepper. Heat a large griddle or two large skillets over medium-high heat. Cook the patties on the griddle until the undersides are well browned, about 2 minutes. Turn and continue cooking until the other sides are browned, about 3 minutes for medium-rare (the patties will feel somewhat soft and barely beginning to firm when pressed on top with a fingertip). During the last 30 seconds of cooking, top each patty with a cheese quarter. Remove from the griddle.
To build each slider, place a lettuce quarter on a roll bottom, then add a patty, about 1 tablespoon each of the onions and sauce, and the roll top. Spear each slider with a skewer (if desired), transfer to a platter, and serve immediately.