For the soup: Coarsely chop 1 of the carrots and put into a food processor. Add onions, celery, peppers, and garlic, pulse until vegetables are finely chopped, then transfer to a large heavy-bottomed pot. Add parsley, salt pork, and 3 quarts water and bring to a boil over medium heat. Reduce heat to medium-low and simmer, covered, for 6 hours.
Using a slotted spoon, transfer salt pork to a food processor, then process until fat liquefies and meat turns into a paste, about 30 seconds. Pass salt pork through a sieve back into pot, using a rubber spatula to press as much paste through the sieve as possible. Skim off and discard fat from broth. Add tomatoes, tomato paste, beef stock base, basil, and oregano to pot. Season to taste with 1 tsp. pepper and simmer over medium-low heat, covered, for 2 hours.
Add 4 cups water to pot, increase heat to medium-high, and bring to a boil. Meanwhile, thinly slice remaining carrots crosswise, then add to pot. Add green beans and peas, reduce heat to medium, and simmer soup, partially covered, until carrots, beans, and peas are soft, about 30 minutes more. Season to taste with salt and pepper.
For the pesto: Put basil, parsley, parmigiano-reggiano, pecorino romano, pine nuts, and oil into a food processor and process until smooth. Season to taste with salt and pepper.
Put soup into a warm tureen, if you like, and spoon pesto into soup. Serve soup in warm bowls garnished with some freshly grated parmigiano-reggiano and pecorino romano, if you like.