In a large stockpot combine the soup vegetables with cold water. Bring to a simmer.
Heat 1/4 cup oil in a saute pan. When hot, add the chicken and smashed garlic and saute until brown on the outside but not cooked through. Add the chicken to the stockpot. Reserve the saute pan.
Simmer the broth for about 1 hour, skimming as needed.
In the reserved chicken pan, saute the chopped garlic, carrots, celery, and onions in 3 tablespoons olive oil until soft, about 5 minutes. Add the butter, white wine, salt, and pepper, to taste. Let simmer an additional 15 to 20 minutes.
Remove chicken from the stock and shred. Strain the stock, discarding the vegetables. Return the broth to the stockpot along with the shredded chicken and sauteed vegetables. Bring to a simmer.
Add the meatballs, escarole, and can of tomatoes to the soup. Simmer until meatballs rise to the top, about 6 to 7 minutes.
Serve with ditalini or tubatini and lots or Parmesean or Romano on top. Serve immediately, or cool and then refrigerate until the next day; reheat before serving.