Peel the carrot and chop finely with the onion and celery (this is known as a soffritto). In a large skillet, add the olive oil and saute the soffritto mixture on low heat until the onion is transparent and the vegetables are soft, about 10-15 minutes. Two minutes before cooking, add the garlic cloves and red pepper flakes, if using and cook for another two minutes.
Add the canned tomatoes along with 1 cup water and season with pepper and salt. Break the canned tomatoes up with the back of your spoon. Simmer uncovered over low heat for a minimum of 20 minutes. Generally speaking, the longer you cook tomato sauce, the better it is. Meanwhile, clean and trim the Italian green beans, cutting them in half if need be to fit better in your skillet. Alternatively, you can cut the Italian green beans into 1 inch long pieces, making them bite size for easier serving.
Add the green beans. Bring the pan back up to a simmer and cook covered until the green beans are tender, about 20 minutes being sure to turn the Italian green beans often so that they cook evenly. Add more water if the tomato sauce is too dry and it starts sticking to the bottom of the pan. When the beans are cooked through, turn the heat off, cover the pan and let rest for 10 minutes.
Test for seasoning and remove the garlic cloves. Chiffonade or gently tear the basil and add to the pan. Serve immediately or serve at room temperature (best for hot summer months!).