How to Make It
Heat canola oil in a large skillet over medium. Add yellow onions and a generous pinch of salt. Cover with a sheet of parchment paper, and cook over medium, stirring every 10 minutes, until very soft and caramelized, about 50 minutes.
Spread chuck and brisket pieces in a single layer on a parchment-lined baking sheet, and freeze until firm, about 30 minutes. Gradually add meat to a meat grinder fitted with a chilled medium plate; grind on medium speed into a chilled bowl. Divide meat into fourths, and shape into balls.
Whisk together ketchup and mayonnaise in a small bowl. Cover and refrigerate until ready to use.
Heat a cast-iron griddle over medium and generously grease with canola oil. Add bread slices and cook until toasted on bottom, about 2 minutes. Flip slices and top each with 1/4 cup Parmigiano-Reggiano. Cook until toasted on opposite side, about 2 minutes. Working quickly, flip bread cheese side down, and cook until cheese is melted and nearly crisp, 1 to 2 minutes. Place toasts, cheese side down, on a baking sheet.
Generously season beef balls with salt. Place balls on griddle, and cook over high until well browned on bottom, about 45 seconds. Turn off heat and, using a large, sturdy spatula, flatten each ball into a 4 1/2-inch patty. Return heat to high, and season patties with salt and pepper. Cook until very crusty on bottom, 2 to 3 minutes. Flip patties, and cook to desired degree of doneness, about 30 seconds for medium-rare. Top each patty with 2 cheddar slices, and transfer to a plate.
Spread plain sides of 4 cheese toasts with ketchup mixture, and season with pepper. Top with patties (cheddar side down), caramelized onions, and red onion. Top with remaining 4 cheese toasts, cheese side up, and serve immediately.