Heat 1 tbsp oil in a large skillet over medium-high. Add onions and peppers and season generously with salt and black pepper. Add butter and sugar and cook, adding a splash or so of water if vegetables are browning too quickly, until golden brown and very tender, 25–30 minutes.
Add onions and peppers to a large bowl, add vinegar and more salt if needed, then set aside.
Heat the remaining 1 tbsp oil in the same skillet, then add meat. Cook for about 4-5 minutes, or until cooked through.
Preheat the broiler and place rolls on a heavy baking sheet. Divide the meat among the four rolls, then top with onions and peppers. Add two slices of cheese to each roll, then place under the broiler for about 2 minutes, or until bubbly. Top with desired pickled vegetables and serve.