Prepare Chicken: Preheat oven to 350°. If chicken has giblets, remove them, and reserve for another use. Rinse chicken, and pat dry. Place chicken in a 13- x 9-inch baking dish, breast side up; brush entire bird with oil, and sprinkle with salt and pepper. Bake at 350° for 1 hour and 10 minutes or until a meat thermometer inserted into thickest portion of thigh registers 165°. Cool 30 minutes. Remove chicken skin and bones, and reserve them for stock. Shred meat with two forks, or coarsely chop. Set aside.
Prepare Dumplings: Whisk eggs in a large bowl. Add melted butter and salt, whisking until smooth. Gradually add flour, stirring until mixture forms a ball. Turn dough out onto a lightly floured surface, and roll into a 12-inch square on a lightly floured surface. (Dough should be about 1/4 inch thick.) Let dough rest, uncovered, 10 minutes. Cut dough into 2-inch squares. Place dumplings on a large parchment paper-lined baking sheet. Cover and chill until ready to cook.
Prepare Stock: Bring 8 cups water and reserved chicken skin and bones to a boil in a 6-qt. Dutch oven. Remove from heat; strain stock into a large bowl. Discard skin and bones. Return stock to Dutch oven. Add onion and next 4 ingredients; bring to a boil. Add dumplings to boiling stock. Reduce heat to medium; cover and cook for 30 minutes, stirring often.
Prepare Roux: Melt butter in a 2-qt. saucepan over medium heat. Whisk in flour until smooth. Cook 10 minutes or until light brown, stirring constantly.
Stir roux and shredded chicken into stock; reduce heat to medium-low, and cook, uncovered, 20 minutes or until thickened, stirring often. Season with additional celery salt, if desired.
Southern Living Off the Eaten Path; Dish, Charlotte, North Carolina