To prepare basil purée, combine basil and 1 garlic clove in a blender; process until smooth. Add water, 2 tablespoons oil, and 1/8 teaspoon salt; process until blended. Set aside.
To prepare vegetables, heat 1 tablespoon oil in a large Dutch oven over low heat. Add bay leaves; cook 1 minute. Add onion, 6 garlic cloves, and thyme; cover and cook 10 minutes. Add carrot, potatoes, 1/2 teaspoon salt, and black pepper; cover and cook 20 minutes. Add squash, bell pepper, and beans; cover and cook 15 minutes. Add the tomatoes; cover and cook 10 minutes. Discard bay leaves and thyme. Spoon stew into individual bowls; drizzle with the basil purée, and sprinkle with Parmesan.