Combine chicken and 5 cups water in a Dutch oven, bring to a boil.
Cover, reduce heat, and simmer 40 minutes or until tender.
Drain, reserving 1 1/2 cups broth.
Skin, bone, and cut chicken into bite-size pieces.
Melt butter in a heavy saucepan over low heat, add 1/4 cup flour, stirring until smooth.
Cook 1 minute, stirring constantly.
Gradually add reserved chicken broth, cook over medium heat, stirring constantly, until mixture is thickened and bubbly.
Add sour cream, half-and-half, 1/2 teaspoon salt, and pepper, stir until smooth.
Stir in chicken, set aside.
Combine 2 1/2 cups flour and 1 1/4 teaspoons salt, cut in shortening with a pastry blender until mixture resembles coarse meal.
Sprinkle cold water (1 tablespoon at a time) evenly over surface, stirring with a fork until dry ingredients are moistened.
Roll two-thirds of dough into a 17- x 13-inch rectangle.
Place in a 12- x 8- x 2-inch baking dish, trim off excess pastry.
Fill pastry with chicken mixture, arrange eggs on top.
Sprinkle with parsley.
Roll remaining one-third pastry to 1-inch thickness, cut into strips.
Arrange strips lattice style over parsley, sealing edges to sides of dish.
Bake at 400° for 35 minutes, let stand 10 minutes before serving.