In a large saucepan, melt butter over medium heat. Add the carrots and sauté until softened, about 5 minutes. Stir in flour, sugar, and salt. Gradually add the chicken broth and cream or half-and-half, stirring to combine.
Bring to a boil; reduce heat and simmer for 15 minutes. Remove from heat and let cool slightly.
Using a blender or food processor, purée the soup until smooth. Return soup to the pan and bring to a simmer.
Drop spoonfuls of dumpling dough into the simmering soup. Cover pan with lid and let cook for 15 minutes.