2 tbsp. olive oil, plus more for drizzling
2 red Italian frying peppers, seeded and thinly sliced
Salt and pepper
A handful of fresh basil, thinly sliced or torn
1 large onion, chopped
4 thick slices beefsteak tomato
1 lb. very thinly sliced boneless rib eye steak (Ask your butcher for a couple of packages to keep frozen—they defrost fast)
4 large eggs
4 soft round Italian rolls or plain kaiser rolls, split
8 slices provolone
1 cup drained giardiniera, finely chopped
In a small skillet, heat 1 tbsp. oil, one turn of the pan, over medium. Add the peppers. Cook, stirring often, until tender, about 10 minutes; season with salt and pepper. Stir in the basil. Transfer to a bowl.
In the same skillet, heat the remaining 1 tbsp. oil, one turn of the pan, over medium. Add the onion. Cook, stirring occasionally, until tender and light caramel in color, 15 to 20 minutes. Add 1/2 cup water; let it absorb into the onion, about 2 minutes. Remove from heat.
Heat a large cast-iron skillet or griddle pan over medium-high. Season the tomato slices; cook until browned in spots, about 1 minute per side. Transfer to a plate.
Increase heat to high. Add a thin drizzle of oil all over the skillet. Working in batches, add the meat (try not to overlap the slices). Toss until browned and crisp, about 3 minutes; season the steak.
In a nonstick skillet, heat a drizzle of oil over medium. Crack in the eggs and cook until fried or over easy, 3 to 4 minutes.
Build sandwiches with the roll bottoms, provolone, onion, tomato, pepper mixture, steak, eggs, giardiniera, and roll tops.