How to Make It
In a large saucepan, cover the chicken wings with 10 cups of water. Add the carrot and celery and bring to a boil. Skim, add a few pinches of salt and simmer over low heat for 1 1/2 hours. Strain the stock and skim off the fat. Measure out 1/2 cup of stock and set it aside.
Heat the 3 tablespoons of olive oil in a large saucepan. Add the scallions and cook over moderate heat until softened, about 3 minutes. Add the onions and cook over low heat, stirring occasionally, until softened, about 8 minutes. Add 2 quarts of the chicken stock and a pinch of salt and simmer over low heat until the onions and scallions are very soft, about 15 minutes. Working in batches, puree the soup in a blender until very smooth. Pour the soup into a clean saucepan and season with salt.
Heat the remaining 2 teaspoons of olive oil in a medium saucepan. Add the potato, garlic and a pinch of salt and cook over low heat, stirring, until the potato pieces are glazed, about 2 minutes. Add the reserved 1/2 cup of chicken stock and 1/2 cup of water and simmer over moderately low heat until most of the liquid has evaporated and the potato is tender, about 10 minutes.
Meanwhile, in a small bowl, whisk the heavy cream until soft peaks form. Pass the potato through a ricer into a medium bowl. Fold the whipped cream into the potato and season with salt.
Ladle the hot soup into warmed shallow bowls. Drizzle 8 drops of balsamic vinegar over each soup, then mound 2 tablespoons of the whipped potato in the center. Sprinkle with cayenne and serve at once.