In a large soup pot, heat the sunflower oil on medium heat. Add the onion, celery, and carrots and cook, covered, stirring occasionally, until soft, about 12 minutes. Add the flour and cook, stirring with a wooden spoon, for 2 minutes more.
Pour in the broth and bring to a boil while whisking constantly. Tie the parsley sprigs, thyme, and bay leaf together with a piece of kitchen twine and add to the soup. Lower the heat and simmer for 15 minutes.
Stir in the chicken and bring to a boil. Remove from the heat.
Whisk the heavy cream, sherry, and salt into the soup and season with pepper to taste. Remove and discard the herb bundle. Divide among soup bowls, sprinkle the top of each soup with the chopped parsley and serve immediately.