Empty a can of tomatoes with their juices into a food processor; pulse until coarsely chopped. Transfer to a bowl and repeat with the remaining two cans of tomatoes (or empty all the tomatoes into a large bowl and break them up with your hands).

Heat the oil in a large (at least 4-quart) saucepan or Dutch oven over medium-low heat. Add the garlic or onion. Cook, stirring often, until the garlic or onion is golden and softened, 5 to 10 minutes for garlic (don't let it burn); about 20 minutes for onion. Stir in the red chile flakes, let them heat for 15 seconds to release their flavor, and then pour in the tomatoes with their juices. Bring to a boil over medium high, stirring frequently, and then reduce the heat to maintain a simmer. Add the salt and basil and simmer, stirring occasionally, until the sauce is full-flavored and has reduced to a thick, saucy consistency, 20 to 40 minutes (some brands of tomatoes take longer to cook down). Pools of orange may appear on the surface, which means the sauce is done. Taste and add the pinch of sugar only if the sauce lacks the natural sweetness of perfectly ripe, fresh tomatoes.

Once the marinara sauce is already made, these variations on the basic sauce can be pulled together in the time it takes to cook the pasta. These sauces will all coat a pound of pasta to serve four.

Basil Marinara: In a small saucepan, heat 3 tablespoons olive oil or butter. Pour in 3 cups marinara, bring to a boil, and then reduce the heat to a simmer. Stir in 1/4 to 1/2 cup torn basil leaves. Simmer for 5 minutes and toss with 1 pound warm pasta. Serve with freshly grated Parmigiano-Reggiano.

Arrabbiata Sauce: In a medium saucepan, heat 2 tablespoons olive oil over medium heat. Add 1/4 pound (3/4 cup) finely diced pancetta or fatty prosciutto, 2 teaspoons minced garlic, and a scant 1/2 teaspoon chile flakes; cook until the garlic is golden, 3 to 4 minutes. Add 3 cups marinara and 1/4 cup torn basil leaves and simmer until hot. Toss with 1 pound warm pasta. Serve with freshly grated Parmigiano-Reggiano.

Mock Bolognese Ragu: In a large saute pan melt 3 tablespoons butter over medium heat. Add 1 cup finely chopped onion, 2 finely chopped cloves garlic, and 1/4 cup each very finely chopped carrot and celery; cook until softened, about 5 minutes. Increase the heat to medium high, add 1 pound ground beef and cook until browned, breaking up the meat with a spoon, 4 to 5 minutes. Pour in 3/4 cup red wine and boil until reduced to 1 tablespoon, 3 to 5 minutes. Add 3 cups marinara and 1/2 cup cream or milk. Simmer until the sauce has thickened enough to softly mound on a spoon, about 8 minutes. Toss with 1 pound warm pasta. Serve with freshly grated Parmigiano-Reggiano.

Pink Sauce: In a large saute pan over medium heat, melt 4 tablespoons butter. Pour in 1-1/2 cups heavy cream. Raise the heat to medium high, simmer, stirring often, until it has reduced to 3/4 to 1 cup, 8 to 10 minutes. Stir in 1/2 cup marinara and season with salt, pepper, and freshly grated nutmeg. Smmer until hot. Toss with 1 pound warm pasta. Serve with freshly grated Parmigiano-Reggiano.

Puttanesca Sauce: In a large saute pan over medium heat, lightly brown 3 large cloves crushed garlic in 3 tablespoons olive oil, mashing the cloves into the oil with a spoon as the cook. Discard the garlic and mash 5 anchovy fillets (rinsed and patted dry) into the hot oil until they're dissolved, about 30 seconds. Stir in 2-1/2 tablespoons rinsed capers and 1/4 teaspoon chile flakes and cook for 3 minutes. Add 3 cups marinara, 1/2 cup pitted, quartered black olives, 1-1/2 teaspoons dried oregano, and 1/4 cup very roughly chopped fresh flat-leaf parsely. Simmer until hot. Toss with 1 pound warm pasta. Serve with freshly grated Romano cheese.