Trim the bulb ends and tough dark green stems off the leeks. Slice the tender white and light green stems in half lengthwise, then soak in cold water for 10–15 minutes to remove dirt. Drain the leeks, rinse well, and slice into thick half-moons.
Heat the 2 Tbsp. of Carapelli Unfiltered Extra Virgin Olive Oil in a large Dutch oven or stockpot over medium-low heat. Add the leeks, garlic, and salt. Cook, stirring occasionally, until the leeks are very soft, about 10 minutes.
Add the potatoes and broth. Cover and increase the heat to medium. Bring the soup to a simmer, then uncover and continue to cook for 30 minutes more.
Puree the soup with an immersion blender until smooth, or carefully ladle into a standing blender to puree (in batches if necessary). Taste and add more salt if needed.
Make the pistou: Pulse the basil, parsley, garlic, pine nuts, salt, and olive oil in a food processor to make a chunky herb sauce. To serve, ladle the soup into bowls and spoon the pistou on top of each.