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.
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.
When chickens are cool enough to handle, pull off skin and slip meat from bones; cut meat into bite size pieces; stir into gravy; pour into an 8 cup baking dish, 8x8x2.
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 8 1/2 inch square; cut into 8 strips, each about one inch wide.
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.)
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.
Bake in hot oven (400°) 30 minutes, or until chicken mixture is bubbly hot, and crust is golden.
Serve immediately.