Heat the olive oil over medium-low to medium heat, add 2 tablespoons of the butter and melt into the oil. When the butter bubbles add the garlic and onions, and season with salt and a little pepper. Cover and cook to soften, stirring occasionally until the onions are very soft and sweet, about 15 minutes, but not brown. Add the stock and tomatoes to the pot, breaking up the tomatoes with a potato masher or wooden spoon. Add several leaves of basil, torn. Gently simmer to thicken, 20 to 30 minutes.
Bring a large pot of water to boil for the pasta, cook to al dente and reserve 1 cup starchy cooking water. Drain the pasta, add back to the pot but keep off the heat. Toss the pasta with the water, remaining 1 tablespoon butter, half the sauce and some cheese. Serve in shallow bowls topped with more sauce, and pass the cheese at table.
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