In a soup pot or Dutch oven over medium heat, warm the olive oil. Add the onion, carrots and celery and cook, stirring often, until softened, about 7 minutes. Stir in the garlic and cook for 1 minute. Add the kale and cook until wilted, about 2 minutes. Add the wine and tomato and bring to a boil. Stir in the stock, water and bay leaf and bring to a boil. Reduce the heat to medium-low, cover partially and simmer until well flavored, about 1 hour. During the last 10 minutes, drain the beans, rinse with cold running water and drain again, then add to the pot along with the basil. Season with salt and red pepper flakes. Discard the bay leaf.
Meanwhile, in a bowl, combine the chicken, half of the cheese, the bread crumbs, egg, the 1/2 teaspoons salt and the 1/4 teaspoons black pepper. Increase the heat under the soup to medium and bring to a light boil. Drop rounded tablespoons of the chicken mixture into the soup. Their texture will firm up on contact with the hot liquid. Cover and cook until the meatballs are cooked through, about 12 minutes. Taste and adjust the seasonings with salt and black pepper.
Ladle into warmed individual bowls and serve hot. Pass the remaining cheese at the table.