Bring 6 quarts of water to a boil and add 2 tablespoons salt.

In a 12 to 14-inch saute pan, place the guanciale slices in a single layer and cook over medium-low heat until most of the fat has been rendered from the meat. Remove the meat to a plate lined with paper towels.

Discard half the fat from the pan, leaving enough to coat, and cook the garlic, onion, and pepper flakes. Place these three ingredients in the pan, add the guanciale, and cook over medium-high heat for 5 minutes, until the onion, garlic, and guanciale are light golden brown. Season with salt and pepper. You may add extra-virgin olive oil if necessary to keep the aromatics from burning. Add the tomato sauce, reduce the heat, and allow to simmer for 10 to 15 minutes.

Cook the bucatini in the boiling water according to the package directions, until tender but still al dente, about 12 minutes. Drain the pasta and add it to simmering sauce. Increase the heat to high and toss to coat. Divide the pasta among 4 warmed pasta bowls. Top with freshly grated pecorino cheese and serve immediately.

In a 3-quart saucepan, heat the olive oil over medium heat. Add the onion and garlic, and cook until soft and light golden brown, about 8 to 10 minutes. Add the thyme and carrot, and cook 5 minutes more, until the carrot is quite soft. Add the tomatoes and juice and bring to a boil, stirring often. Lower the heat and simmer for 30 minutes until as thick as hot cereal. Season with salt and serve. This sauce holds 1 week in the refrigerator or up to 6 months in the freezer.