To a large saucepan, add the chicken, 1 teaspoon salt, and enough water to cover. Add the whole garlic clove, the parsley stems, half the celery, half the carrots, and half the onions. Bring to a boil over high heat, skim, then turn down to low and simmer until the chicken is tender and can be pulled off the bone, about 1 hour.
Transfer the chicken to a large bowl and let cool. Strain and skim any fat from the broth. When the chicken is cool enough to handle, use two forks to shred the meat off the bones into bite-sized pieces – the bones have given up all their goodness now and can be discarded. Cover the shredded chicken until needed. (This will give you more chicken than is necessary for this dish. Either go to town and have your soup chock-full of chicken or save some of the meat for sandwiches or salads.)
In another large saucepan over medium-low heat, warm the olive oil and cook the remaining celery, carrot, onion, and garlic, the sliced leeks, and the remaining 1/2 teaspoon of salt until soft, 10 to 15 minutes.
Pour in the reserved chicken broth, increase the heat to medium and bring to a simmer. Add the risoni or other pasta and cook, according to the instructions on the packet, until al dente.
Stir in the shredded chicken, baby spinach, and parsley leaves. Taste and adjust seasoning, if necessary. Serve hot, with freshly ground black pepper and crusty bread.