In a large bowl, gently mix the short rib and short plate with the cooled goose fat until just combined. Season lightly with kosher salt and black pepper. Using a 4-inch/10 cm ring mold, form 6 even patties (do not compact). Refrigerate 15 to 20 minutes to firm.
Stir together the mayonnaise, Dijon, confit garlic, and cornichons until creamy. Cover and refrigerate.
Set a medium skillet over medium heat. Add the onions and cook, stirring, until deep golden, 10 to 12 minutes. Add the whiskey and let it bubble for 30 seconds. Stir in the barbecue sauce and cook over low heat until glossy and caramelized, 5 to 7 minutes. Keep warm.
Heat a cast-iron skillet or grill pan over medium-high until very hot. Sear the patties until well browned, about 3 to 4 minutes per side for medium (130° to 135°F/54° to 57°C); adjust to your preferred doneness. Let rest for 3 minutes.
To assemble, spread sauce generously on the bottom bun, add a patty, spoon on a thick layer of onion relish, and cap with the top bun (spread with more sauce, if you like). Serve immediately.