For the smoked chicken broth: Place the chicken stock, garlic, celery, carrot, onion and smoked chicken carcass and backbone into a large stockpot or Dutch oven. Simmer gently for 1 hour over a low flame. Strain, transfer to a large container and reserve (cool completely if making ahead of time).
For the minestrone: Add the oil to a Dutch oven and bring to medium heat. Sweat the carrots, celery and onions until softened, 3 to 5 minutes. Add the garlic and sweat until fragrant, 30 seconds. Add the potatoes, zucchini, cabbage and season with salt and pepper. Sweat the vegetables slowly until all are crisp-tender, about 8 minutes.
Pour in the reserved stock and the tomatoes. Add the Parmesan rind and simmer 12 to 15 minutes; the vegetables should be tender. Remove from the heat. Add the navy beans and shredded chicken and stir to combine. Season with salt and pepper.
Ladle the minestrone into deep bowls and garnish with the shaved parmesan and chopped parsley and basil. Serve immediately.
Preheat the oven to 350 degrees F.
For the chicken: Using kitchen shears, remove the backbone of the chicken and set aside. Cut the chicken in half, straight through the breastbone. By separating the chicken into two pieces it will ensure that the chicken gets smoked more thoroughly. Sprinkle the chicken pieces with salt, pepper, the granulated garlic and smoked paprika.
Place 1 deep aluminum roasting pan over the stove top. Line with aluminum foil. Add the soaked wood chips to a large piece of foil, and then fold in the sides to make a pouch. Punch holes in the top of the pouch, set it directly over a gas burner and turn on high. Leave for 2 to 3 minutes, watching carefully. Once it begins to smoke, using tongs, transfer the pouch to the bottom of the lined roasting pan. Set a wire rack over the chips in the lined roasting pan and position it so the wood chips are directly over the burner and continue to smoke. Place the chicken pieces bone-side down on the rack. Cover tightly with aluminum foil to seal in the smoke. Put the roasting pan into the oven and smoke until the internal temperature reaches 165 degrees F on an instant-read thermometer, 45 to 60 minutes. Remove and cool. Shred the smoked chicken breast.