1. Vegetable Stock: Heat oil in a large pot over medium. Add leeks, carrots, and celery; cook, stirring occasionally, until softened, about 10 minutes. Add 10 cups water, thyme, basil, parsley, bay leaf, and peppercorns. Bring to a boil, then reduce heat to a gentle simmer and cook, uncovered, 1 hour. Season with salt.

2. Strain stock through a fine sieve into a medium pot, pressing on solids to extract as much liquid as possible (you should have 7 cups); discard solids.

3. Soup: Prepare ice-water bath. Bring a large pot of salted water to a boil. Add fava beans and blanch 2 minutes. Transfer to ice-water bath; drain. Remove outer skin from fava beans.

4. Bring stock to a simmer over medium-high heat. Add lima beans, tomatoes, and pasta; cook, stirring occasionally, until beans are tender and pasta is al dente, about 10 minutes. Stir in fava beans; season with salt and pepper. Ladle soup into bowls; top with basil, cheese, and a drizzle of oil.