['2 tablespoons olive oil', '4 large onions, sliced 1/4-inch thick', '3 large red and/or green bell peppers, seeds removed, sliced 1/4-inch thick', 'Kosher salt, freshly ground pepper', '3 tablespoons white or red wine vinegar', '4 pounds boneless beef short ribs', '8 (10-inch-long) Italian sub rolls', 'Vegetable oil (for griddle)', 'Kosher salt', 'Freshly ground pepper', '10 ounces white or yellow American cheese slices (about 16)', '10 ounces mild or sharp provolone slices (about 16)', 'Ketchup and/or hot sauce (for serving; optional)']
Heat oil in a large pot over medium-high. Add onions and bell peppers and season generously with salt and black pepper. Cook, stirring often at first then only occasionally as vegetables soften, adjusting heat as needed and adding a splash or so of water if vegetables are browning too quickly, until golden brown and very tender, 25–30 minutes.
Remove pot from heat, add vinegar, and give everything a good stir. Taste and season with more salt if needed.
Place short ribs on a parchment-lined rimmed baking sheet and freeze, uncovered, until firm around the edges but still yielding in the center, 45–60 minutes.
Cut short ribs in half crosswise, then lengthwise into very thin slices. (It’s better to have thinner, less-regular slices than thick, even slices. Either way, don’t stress; whatever you do will be more than fine.) Divide meat into 8 portions (about 8 oz. each), using sheets of parchment paper or plastic wrap as dividers.
Slice bread in half lengthwise, keeping a long side intact. Heat a large griddle across 2 burners over medium—a flick of water should sizzle on contact. (Or use a large cast-iron skillet and cook 1 portion at a time.) Lightly coat griddle with oil. Place 2 portions of meat (about 1 lb.) on griddle; season generously with salt and pepper. Cook, pressing down and pulling slices apart using 2 metal spatulas, until halfway cooked through (you should still see some raw spots). Scoot meat into 2 oblong portions no more than 1" high and top with some onion–bell pepper mixture. Top with 2 slices American cheese and 2 slices provolone cheese per pile. Cook, undisturbed, until meat is browned and crisp underneath and cheese is melted, about 4 minutes. Transfer each portion to a roll; slice in half crosswise. Repeat with remaining ingredients. Serve topped as desired.
Vegetables can be cooked 3 days ahead. Let cool, then cover and chill. Reheat before assembling sandwiches. Meat can be sliced 1 day ahead (it will start to oxidize after that); cover tightly and chill, or freeze up to 1 week and thaw before cooking. Sandwiches can be made 20 minutes ahead; immediately wrap in parchment paper, then foil.
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