1. In 5-quart Dutch oven or saucepot over high heat, heat chickens, pepper, marjoram, 4 cups water, and 2 teaspoons salt to boiling. Reduce heat to low; cover and simmer 35 minutes or until chicken is fork-tender.
2. When chicken is done, reserve 1 cup broth. Cool chicken until easy to handle; remove and discard bones and skin; cut meat into 1-inch pieces.
3. In 3-quart saucepan over high heat, heat carrots, celery, onions, and reserved 1 cup chicken broth to boiling. Reduce heat to low; cover and simmer 10 minutes or until vegetables are almost tender. Remove from heat. With slotted spoon, remove vegetables to medium-sized bowl.
4. In small bowl, blend 1/3 cup flour with 3/4 cup half-and-half; gradually stir into broth in saucepan; stir in remaining half-and-half. Over low heat, cook, stirring until sauce thickens slightly. Stir in chicken, cooked vegetables, lima beans, mushrooms, and 1 1/2 teaspoons salt. Spoon mixture into 13" by 9" baking dish.
5. Prepare pastry: In medium-sized bowl, stir 3 cups all-purpose flour and 1 1/2 teaspoons salt. With pastry blender or two knives used scissors fashion, cut shortening into flour to resemble coarse crumbs. Sprinkle about 1/2 cup water, a tablespoon at a time, into flour mixture, mixing lightly with a fork until pastry is just moist enough to hold together; shape pastry into a ball.
6. On lightly floured surface with floured rolling pin, roll two-thirds of dough into 15" by 11" rectangle. With knife, cut several small slits in center of pastry. Place pastry loosely over chicken mixture. Trim pastry edge, leaving 1-inch overhang; fold overhang under and press gently all around baking dish to make a high stand-up edge. Roll and cut remaining dough to make decorative design for top of pie.
7. In cup, beat egg yolk with 1 teaspoon water. Brush pastry with egg-yolk mixture. Bake pie in 350°F. oven 1 hour or until crust is golden and chicken mixture is heated through.