Place chicken in large kettle and cover with water. (Approx. 2 quarts) Bring to a boil; reduce heat to low and simmer for 1 hour.
Heat oil in separate pan and saute onions for about 5 minutes and set aside.
When chicken is done cooking, remove from broth, cool and dice. Set aside.
Add bay leaves, dill, sauteed onion, and carrot to broth and simmer for 1 hour.
Remove carrot and add rice. (If you prefer, you can slice the carrot and add it back in the soup.)
In a small bowl, beat the lemon juice and eggs together. Pour two cups of broth (from soup) slowly into the bowl, whisking constantly. Add the mixture back into the pot of chicken soup and stir until well blended.
Add salt and pepper and serve hot.