1. In a 5 quart soup pot, heat oil. Add onion. Cook and stir until onion is tender. Add squash, broth, 1/2 tsp. salt and 1/2 tsp. pepper. Cover and bring to a boil. Reduce heat and simmer 15 minutes or until squash is tender.
2. Meanwhile, in a small bowl using a fork, mash and stir goat cheese, egg, flour, 1/4 cup Parmigiano-Reggiano cheese, chives, 1/2 tsp. salt and 1/4 tsp. pepper until completely combined. Cover and refrigerate.
3. Using an immersion blender or traditional blender (using enough stock to allow the machine to do its work), purée soup until smooth.
4. Return soup to a boil then reduce to a simmer. Using a teaspoon, drop small balls of the dumpling mixture into the soup. Cook for 2 minutes once all dough has been added. Serve sprinkled with additional Parmigiano-Reggiano cheese.