Bring chicken and water to cover to a boil in a large Dutch oven; reduce heat, and simmer 1 hour. Remove chicken; cool. Pour broth through a wire-mesh strainer into a large saucepan, discarding solids. Skim off fat. Return broth to Dutch oven; bring to a simmer.
Skin and bone chicken; cut into bite-size pieces. Add chicken, bouillon granules, and pepper to broth; return to a simmer.
Combine flour, baking powder, and salt in a medium bowl. Cut in shortening with a pastry blender until mixture is crumbly. Add milk, stirring until dry ingredients are moistened.
Turn dough out onto a lightly floured surface. Roll out to 1/8-inch thickness; sprinkle lightly with flour, and cut into 3- x 2-inch strips.
Bring broth mixture to a boil. Drop strips, 1 at a time, into boiling broth, stirring gently, until all are added. Reduce heat, and simmer, stirring often, 20 minutes. Stir in egg just before serving.