1. Prepare Chicken and Chicken Stock: Bring chicken, next 6 ingredients and water to cover to a boil in a Dutch oven over medium heat. Reduce heat to medium-low, and simmer 45 minutes or until chicken is done. Remove chicken, reserving liquid and vegetables in Dutch oven. Cool chicken 30 minutes; skin, bone, and shred chicken. Reserve bones; loosely cover chicken, and refrigerate until ready to use.
2. Place reserved chicken bones into Dutch oven with reserved liquid and vegetables, and cook over medium heat 1 hour. Pour liquid through a wire-mesh strainer into a 3-qt. saucepan, discarding solids. Reserve 1 1/2 qt. chicken stock in saucepan.
3. Prepare Vegetables: Bring reserved stock to a boil over medium-high heat. Add turnips, carrots, and mushrooms, and cook 3 to 4 minutes or until crisp-tender. Remove from stock. Add red pearl onions and next two ingredients to stock, and cook 7 to 8 minutes or until crisp-tender. Remove onion mixture from stock, reserving 4 cups stock.
4. Prepare Chicken Sauce: Melt butter in a heavy 2-qt. saucepan over medium heat. Reduce heat to low, whisk in flour, and cook, stirring often, 10 minutes or until golden. Increase heat to medium-high, and slowly whisk in 4 cups reserved chicken stock until combined. Cook, stirring often, 15 minutes or until thoroughly heated.
5. Prepare Dumplings: Pulse flour and next 4 ingredients in a food processor 4 to 5 times or until crumbly. Transfer to a large bowl, and stir in buttermilk and next 3 ingredients, stirring until well blended.
6. Assemble Chicken and Dumplings: Preheat oven to 400°. Place shredded chicken and vegetables in a 4-qt. baking dish. Pour Chicken Sauce over chicken and vegetables; top with bacon and eggs.
7. Spoon batter by rounded tablespoonfuls over chicken mixture, gently pressing dumplings down into sauce. Bake at 400° for 25 to 30 minutes or until mixture is bubbly and dumplings are golden brown. Let stand 5 minutes.