Bring the leeks, carrots, onion, celery, potato, zucchini, spinach, tomatoes, water, cheese rind, and salt to a boil in a large stockpot or Dutch oven. Reduce the heat to medium-low and simmer, uncovered, stirring occasionally, until the vegetables are tender but still hold their shape, about 1 hour. Remove and discard the cheese rind.
Add the beans and cook just until heated through, about 5 minutes. Remove the pot from the heat. Stir in the pesto. Adjust the seasonings, adding pepper and more salt, if necessary. Serve immediately.