In a large heavy skillet over medium heat, cook the diced guanciale, stirring often until most of the fat is rendered and the pork is crispy. With a slotted spoon remove the pork to a separate dish and set it aside. Leave the rendered fat in the pan.
Reduce the heat slightly and add the anchovy fillets to the pan. Stir frequently for a few minutes until the anchovy virtually disappears or melts into the oil. Add the diced onion and cook, stirring frequently until the onion is soft, about 5 minutes. Add the garlic and sauce for another minute or so. Ad the tomato paste and red pepper flakes and stir to incorporate. Add the tomatoes to the pan, scraping up any browned bits of meat or and onion that have stuck to the bottom of the pan. Add the guanciale to the space and bring it all to a low simmer.
Let the sauce simmer gently for about 40 minutes, until the flavors are well incorporated.
Cook the pasta until al dente, according to the directions on the package. Drain the pasta, reserving about a cup of pasta cooking water. Add the cooked pasta to the pan with the sauce and, over low to medium heat, stir and toss the pasta to incorporate the sauce. Add the cheese and basil and toss to combine those. Then garnish with some more basil. Pass the extra cheese at the table for diners to add more as they choose.