1. In a pot, bring water and chicken to boil. Remove all scum. Reduce heat, cover and let simmer for about 30 minutes.
2. Remove meat and set aside to cool slightly. Then cut into bite-size pieces. Return to pot.
3. Add potatoes, cover and simmer for 10 minutes.
4. Meanwhile, heat oil in a skillet over medium heat. Saute onions and carrots for about 5 minutes, or until onions become soft and translucent.
5. Make the dumplings. In a small bowl mix together flour, egg and milk (add more milk if necessary). Season with salt and pepper, mix well.
6. Add carrots, onions and bay leaf to the pot. Drop the dumpling dough by half-teaspoonfuls into the simmering soup. Cover and simmer for 10 more minutes.
7. Season with salt and pepper to taste, sprinkle with dill, turn off the heat, remove the bay leaf and let the soup stand for a few minutes before serving.