In a large flameproof casserole or Dutch oven with a snug lid, heat the oil with the bay leaves over low heat. When fragrant, add the onions, 6 of the garlic cloves (if using canned beans instead of fresh, add all 7 cloves), 2 of the thyme sprigs (if using canned beans instead of fresh, add all 3 sprigs), and the sage, stirring to coat everything thoroughly with oil. Cover and cook over low heat as you prepare the rest of the vegetables. Leave very small carrots whole and unpeeled; if using larger ones, peel them and cut them into 2-inch lengths. Add them to the pot. If the potatoes are the size of large marbles, leave them whole, but quarter larger ones or cut fingerlings in half lengthwise. Add the potatoes to the pot in one layer; season with salt and pepper. Add the wax beans, bell peppers, and summer squash to the pot in layers, seasoning each layer with a little salt and pepper as you go. Add the tomatoes, sprinkling their juices over all. Cover and cook over low heat until the vegetables are tender, 40 to 65 min. If tightly covered, the vegetables will produce plenty of flavorful juices. There's no need to stir, but if the pot seems dry, add a few tablespoons of water or dry white wine, if you like.
Make the pesto: In a blender, process the basil and garlic with the oil, adding a little water to loosen if needed. Add the salt and the cheese, if using. Taste and adjust the seasonings if needed.
Cook the shell beans: If you're using fresh shell beans, put them in a saucepan with enough water to total 3 cups, beans included. Add the remaining garlic clove, thyme sprig, and a little olive oil. Simmer uncovered until tender, 30 to 45 min. Season with salt and pepper. Add the beans and their cooking liquid to the pot (if using canned beans, add a bit of water or stock). Ladle into soup plates, drizzle some pesto over each dish, and serve.