Put chicken broth, turkey, onion halves, carrot halves, 1 celery stalk, and 1 bay leaf in a large stock pot. Bring to a boil, then simmer about 1 1/2 hours. Finely dice the remaining onion, carrot and celery and reserve.
Dice the turkey meat. Make sure meat pieces are no larger than the size of a soup spoon. (If preparing soup the next day, be sure to store leftover turkey meat in an airtight container before placing it in the refrigerator, top with 1 or 2 ladles full of broth to keep meat moist.)
Before straining broth, remove large bones and carcass with tongs. Strain the broth through a sieve, covered with wet cheese cloth. Discard the solids. Transfer broth to a bowl set in a bath of ice water, which will cool the broth quickly and help keep it fresher longer. This can be done the night before and stored in the refrigerator until the next day.
In a large soup pot, heat garlic cloves in the olive oil. Allow to brown slightly and add minced carrots, celery, and onion. Sweat over medium-low heat until softened, about 7 or 8 minutes.
Dice the leftover vegetables (here Brussels sprouts, sweet potatoes, and green beans). Add the chopped sage to the soup pot along with the turkey broth and the remaining bay leaf. Bring to a simmer. When simmering, add Brussels sprouts, green beans and diced turkey meat to the soup. Bring it back up to a simmer. Finally, add the sweet potatoes to the center, and gently push them down. Turn the heat off and cover. Allow to sit and steam for 5 to 7 minutes.
Let simmer for 5 more minutes and serve.
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