How to Make It
In a large bowl, combine the flour and a generous pinch of salt. Sprinkle the milk and 2 tablespoons of the olive oil on top and, using a fork, mix to form a shaggy dough. Knead the dough lightly until evenly moistened and then, using water to keep your hands moist, roll teaspoons of the shaggy dough between your palms to form 2-inch-long dumplings, a scant 1/4-inch thick. Transfer the dumplings to a parchment paper–lined baking sheet.
In a large saucepan, heat the remaining 2 tablespoons of olive oil. Add the leek, carrot, frisée and a generous pinch of salt and cook over moderate heat, stirring occasionally, until just softened, about 7 minutes. Add the chicken stock and bring to a boil, then simmer over moderately low heat until the vegetables are tender, about 5 minutes. Return the soup to a boil, add the dumplings and simmer over moderate heat until the dumplings are cooked through, 3 to 5 minutes. Season with salt and pepper.
Ladle the dumplings, broth and vegetables into shallow bowls. Drizzle olive oil on top, garnish with shaved Parmigiano cheese and serve.