1. Place chickens in a large heavy kettle or Dutch oven; add 2 cups water, salt, pepper and carrots. Heat to boiling; reduce heat; cover; simmer 45 minutes. Add peas; simmer 15 minutes longer, or until chicken is tender. Remove chicken to a bowl to cool.
2. Skim fat from chicken broth-vegetable mixture; reserve 2 tablespoons fat. Melt butter or margarine with reserved chicken fat in a medium-size saucepan; stir in flour; cook, stirring constantly, just until bubbly. Stir in chicken broth-vegetable mixture; continue cooking and stirring until gravy thickens and bubbles 1 minute.
3. When chickens are cool enough to handle, pull oft" skin and slip meat from bones; cut meat into bite-size pieces; stir into gravy; pour into an 8-cup baking dish, 8x8x2.
4. Combine biscuit mix and sour cream in a small bowl; stir to form a stiff dough; turn out onto a lightly floured board; knead a few times; roll out dough to 1/4,-inch thickness ; trim to make an 81/2-inch square; cut into 8 strips, each about one inch wide.
5. Using 4 of the strips, make a lattice design on top of the chicken mixture, spacing evenly and attaching ends firmly to edges of the dish. Place remaining strips, one at a time, on edges of dish, pinching dough to make a stand-up rim; flute rim. (Or, roll out dough to a 9-inch square and place over chicken mixture; turn edges under, flush with rim; flute to make a stand-up rim. Cut slits near center to let steam escape.)
6. Combine egg with 1 tablespoon water in a cup; mix with a fork until well blended; brush mixture over strips and rim; sprinkle with sesame seeds.
7. Bake in hot oven (400°) 30 minutes, or until chicken mixture is bubbly-hot, and crust is golden.