In a large pot, warm oil over medium heat. Add onion, carrots, celery, 1 1/2 teaspoon thyme, salt and pepper. Sauté for 6 minutes or until vegetables are softened. Add the flour and cook, stirring for another 2 minutes. Whisk in the chicken broth and bring to a simmer. Turn the heat to low and cook for 10 minutes.
While the broth is simmering, prepare the dumplings. Combine the milk and egg in a medium bowl. Combine the flour, baking powder, 2 tablespoons parsley, remaining 1/2 teaspoon of thyme and 1/2 teaspoon salt together in another bowl. Add flour mixture to milk mixture, stirring with a fork just until dry ingredients are moistened.
Add the chicken and the cream to the broth and veggies, and bring back to a simmer.
Drop dumpling batter by heaping tablespoons onto simmering chicken mixture. Cover and cook for 8 to 10 minutes or until the dumplings are done (do not allow chicken mixture to boil).