How to Make It
Heat 2 tablespoons of the olive oil in a large saucepan. Add the onion and garlic and cook over moderate heat, stirring frequently, until lightly browned, about 5 minutes. Add the tomatoes, stock, water and cranberry beans and bring to a boil. Cover partially and cook over moderately low heat until the cranberry beans are tender, about 15 minutes. Season with salt and pepper. Add the string beans and flat beans, cover partially and cook over moderate heat until tender, 10 to 12 minutes.
Meanwhile, bring a medium saucepan of salted water to a boil. Add the pasta and cook, stirring, until al dente. Drain the pasta and add it to the beans along with the parsley, vinegar, 1/2 cup of Parmesan and the remaining 3 tablespoons of olive oil. Season with salt and pepper. Serve the minestra in deep bowls and pass extra Parmesan at the table.