In a deep pot (ideally an earthenware one you can bring to the table for easy serving), add the stock, sake, soy sauce, and salt. Bring to a boil, mix well, then add the chicken and vegetables and cook over medium heat until halfway done.
Add the rice and cook, stirring occasionally, until the chicken and vegetables are completely cooked. Add the beaten eggs, stir once and turn the heat off. Cover the pot for a minute or two-the egg will still be undercooked and runny at this point-then sprinkle the top with either the green onions or trefoil, mixing it into the porridge if desired.
Serve immediately in soup bowls. A plate of tsukemono (pickles) would provide the perfect accompaniment to this dish.