Prep: 45 minutes; Cook: 1 hour; Bake: 45 minutes.
1. Rinse hen under running water, remove giblets, and place hen in a Dutch oven with enough water to cover. Add 1 teaspoon salt and 1/2 teaspoon pepper, and bring to a boil over high heat. Reduce heat to medium-low, cover, and simmer 1 hour. Remove hen from pot, and keep liquid warm. Pull meat from bone once hen cools, and discard carcass. Tear meat into small pieces (about 3 cups).
2. Preheat oven to 350°. Crumble Cornbread into a large bowl. Add onion and celery, sage, remaining 1 1/4 teaspoons salt, and remaining 1 teaspoon pepper. Bring reserved cooking liquid to a boil, and stir 4 cups into mixture, 1 cup at a time. Dressing should be loose but not gummy. Add more cooking liquid as needed. Lightly stir in chicken and egg.
3. Spoon dressing into a 13- x 9-inch baking dish, and bake at 350° for 45 minutes or until heated through and slightly browned on top. To make ahead, cover dressing and refrigerate up to 2 days before baking.