Mince 2 cloves garlic and set aside. Sort and rinse adzuki beans as described in the Basic Beans recipe. Chop onion and cut carrot into 1/4 inch wide moons. Put garlic, beans, onion, salt, carrot, and water into a medium pot. Cover pot and bring to a boil over high heat, then reduce heat to low so bean mixture barely boils. Cook until beans are tender, about two hours, then make sauce.