To prepare pistou, combine 1/2 teaspoon salt and 2 garlic cloves in a food processor; process until minced. Add basil; process until finely chopped. Add 1 1/2 tablespoons oil and lemon juice, process until blended. Set aside.
Preheat oven to 400°.
To prepare soup, combine tomatoes, 2 teaspoons garlic, 1/8 teaspoon salt, and 1/8 teaspoon pepper in a small bowl; toss well. Coat a foil-lined baking sheet with cooking spray. Spoon tomato mixture onto baking sheet. Bake at 400° for 15 minutes. Set aside.
Heat oil in a Dutch oven over medium heat; add 4 teaspoons garlic and onions. Sauté 10 minutes, stirring frequently; add remaining 1/8 teaspoons of salt and pepper. Add sweet potato, carrots, thyme, and bay leaf; sauté 10 minutes, stirring frequently. Add white beans; cook 5 minutes, stirring occasionally. Add broth, water, and green beans; simmer 15 minutes or until beans are tender. Discard bay leaf and thyme sprigs. Stir in pistou and tomato mixture.