In a 10-inch Dutch oven or stockpan, heat the olive oil over a medium-high flame until hot but not smoking. Add the onion, celery, and leek and cook, stirring occasionally, until soft and light golden brown, about 5 minutes. Add the faro, beans, and tomato paste, stirring so that the tomato paste is spread throughout the pan. Add salt and pepper, to taste. Add water until the mixture is completely covered. Let the liquid come to a boil then lower the heat and let the soup simmer gently for 30 minutes.
Add the peas and the carrots, stirring to combine, and continue to cook for another 30 minutes, adding more water when necessary to keep the soup from getting to dry.
To serve: Divide evenly between 4 warmed soup bowls and garnish, to taste, with basil and Parmigiano-Reggiano.