Heat a large Dutch oven over medium heat. Add 2 teaspoons oil; swirl to coat. Add leeks, celery, 2 teaspoons garlic, and salt; sauté 5 minutes. Add stock or water and thyme. Cover; bring to a boil. Add sno peas, green beans and cannellini beans, and simmer, uncovered, 4 minutes. Discard thyme. Stir in 1 cup spinach.
Place 1 cup spinach, basil, cheese, zest, juice, 1 teaspoon garlic, and 2 teaspoons oil in a food processor; process until smooth. Divide soup among 4 bowls; top with pesto.
Nutrition information based on using water (not stock).