Preheat the oven to 350 degrees .
Bring a large pot of water to a boil for the spaghetti.
Heat a medium cast-iron skillet over medium-high. Coat the steaks with a fat drizzle of olive oil, sprinkle with the rosemary and a third of the garlic; season with kosher salt and coarsely ground black pepper. Add the steaks to the skillet and cook until browned on both sides, 3 to 4 minutes per side. Using tongs, tilt the steaks to brown the edges. Transfer the skillet to the oven and roast until an instant-read thermometer inserted into the centers of the steaks registers 135 degrees , 10 to 12 minutes. Let rest on a carving board.
Meanwhile, in a large skillet, heat 2 tbsp. oil, two turns of the pan, over medium-high. Add the peppers, chiles, onion, oregano and the remaining garlic; season. Cook, stirring often, until vegetables are softened, 5 to 7 minutes. Add the tomato paste and cook, stirring constantly, until dark red, about 1 minute. Add the wine; cook, stirring constantly, until evaporated, about 1 minute. Add the stock to the sauce; simmer, stirring often, until reduced by half, about 8 minutes. Reduce the heat to low.
Salt the boiling water, add the spaghetti and cook to al dente. Add half a mug of the starchy cooking water to the sauce. Drain the spaghetti. Add the spaghetti, grated cheese and parsley to the sauce; toss to coat. Drizzle with EVOO.
Slice the steaks 1/4 inch thick, divide among 4 plates and top with some greens and the shaved cheese; douse with the juice of the halved lemon. Serve the spaghetti alongside the sliced steak or in separate small, shallow bowls.