Bring broth and dill to a boil in a large saucepan. Add orzo. Reduce heat, and simmer 5 minutes or until orzo is slightly tender. Remove from heat.
Place eggs and juice in a blender; process until smooth. Remove 1 cup broth from pan with a ladle, making sure to leave out orzo. With blender on, slowly add broth; process until smooth.
Add carrot, salt, pepper, and chicken to pan. Bring to a simmer over medium-low heat, and cook 5 minutes or until chicken and orzo are done. Reduce heat to low. Slowly stir in egg mixture; cook 30 seconds, stirring constantly (do not boil).