How to Make It
Set a rack over a rimmed baking sheet. In a bowl, toss the beef chuck with 2 teaspoons of salt to evenly coat. Spread in a single layer on half of the rack. Add the short rib to the bowl and toss with 1 teaspoon of salt to evenly coat. Spread in a single layer alongside the chuck. Refrigerate the meat uncovered for 8 hours. Store the meat grinder attachment—including the blade and coarse grinding plate—in the freezer.
In a nonstick skillet, heat 2 tablespoons of the oil. Add the onions and cook over moderate heat, stirring frequently, until deep golden, 25 minutes. Season with salt and keep warm.
Working over a large bowl, gently pass the short rib through the grinder. Gently press the beef chuck through the grinder, gradually adding the ground short rib as you go so they are ground together. Gently shape the ground beef into ten 1-inch-thick patties, packing them just enough so they hold together.
Preheat the oven to 425°. Arrange the English muffins on 2 baking sheets. Set a cheese slice on each top half. Toast until the muffins are crusty and the cheese is melted, about 5 minutes. Spread some caper mayonnaise on the bottom halves.
Preheat a cast-iron griddle. Add 2 tablespoons of the canola oil and cook half of the burgers over moderately high heat until browned, 2 to 3 minutes. Flip the burgers and top with half of the caramelized onions, lightly pressing them into the burger. Cook until browned and pink within, about 2 minutes for medium-rare. Repeat with the remaining 2 tablespoons of canola oil, burgers and caramelized onions. Set the onion burgers on the muffins and serve.