1. Heat broth and chicken in a large saucepan until boiling. Reduce heat to low; cover and simmer 2 minutes.
2. With a slotted spoon, remove chicken to a 2-quart casserole dish and set aside.
3. Add carrots to hot broth and simmer about 5 minutes, until fork-tender, then remove and add to the cooked chicken. Reserve the broth and set aside.
4. In another medium saucepan, melt butter or margarine over medium heat. Add the onion and cook until translucent, about 5 minutes.
5. Add the flour and cook for another 5 minutes, stirring constantly. Remove from heat and slowly whisk in the reserved broth. Return to heat and cook, stirring until sauce boils and thickens.
6. Remove from heat; blend in milk and salt, then pour sauce over chicken and carrots.
7. Mix in the peas and parsley, and set aside.
8. Preheat oven to 375° F.
9. In a small bowl, combine the flour, cornmeal, sugar, and baking powder. Using a fork, blend in the milk, egg, and oil.
10. Pour over the chicken filling and spread to edges of dish with a spatula.
11. Bake 45 to 50 minutes if 1 large pie (40 minutes if in 2 smaller casseroles; 35 minutes if 9 individual ones), or until crust is golden and filling is bubbly. Let stand 10 minutes before serving.
(To reheat from frozen, double the baking time.)