We love the combination of wild rice and hominy in this simple soup. Brown rice or any leftover cooked grain would work well, too. We use a few tricks to get maximum comforting flavor into the soup, starting by enriching the chicken broth with a roast chicken carcass and by adding chopped cilantro and scallion at the very end for freshness.
Cook the rice according to package directions; set aside to cool.
Put the chicken carcass and bones in a large pot (they can be in pieces). Cover with the broth and 4 cups water. Bring to a boil over medium-high heat; reduce the heat and simmer for 20 minutes. Skim any foam or fat from the top of the broth with a ladle as necessary.
Remove the bones and carcass with tongs or a slotted spoon; set aside to cool. Add the carrots, celery, onion and bay leaf to the broth. Bring back to a simmer and cook until the vegetables are about half cooked (they will still offer a little resistance when poked with a knife), about 10 minutes.
Meanwhile, when the carcass and bones are cool enough to handle, pick off any meat and shred it into bite-size pieces. Roughly chop the cilantro and scallions together.
Add the shredded chicken, cooked rice, hominy and cilantro-scallion mixture to the broth and simmer until warmed through, about 1 minute. Season with 1/2 teaspoon salt or more to taste. Serve hot with lime wedges.