1. Remove giblets and neck from chicken; trim excess fat. Place chicken, 10 cups water, and next 7 ingredients (through parsley bunch) in an 8-quart stockpot. Bring to a boil; reduce heat and simmer, uncovered, 1 hour, skimming foam from the surface as necessary. Remove from heat, and let stand for 20 minutes.
2. Remove skin from chicken; remove chicken from bones, discarding skin and bones. Shred chicken with 2 forks. Strain stock through a sieve into a large bowl; discard solids. Cool stock and chicken to room temperature. Cover and chill stock and chicken separately for 8 to 24 hours. Skim solidified fat from surface of stock; discard fat. Reserve 8 cups stock and chicken. Refrigerate remaining stock in an airtight container for up to 1 week, or freeze for up to 3 months.
3. Melt 1 tablespoon butter in a Dutch oven over medium heat. Add 1 cup onion, 1/4 cup celery, and sliced carrot to pan; cook 8 minutes, stirring occasionally. Stir in 1 tablespoon flour, 3/4 teaspoon salt, and ground pepper; cook, stirring constantly, 1 minute. Stir in 8 cups reserved stock. Bring to a boil; reduce heat and simmer 15 minutes, stirring occasionally.
4. Weigh or lightly spoon 4.5 ounces flour (about 1 cup) into a dry measuring cup; level with a knife. Combine flour, baking soda, and remaining 1/4 teaspoon salt, stirring with a whisk. Cut in remaining 2 tablespoons butter with a pastry blender or 2 knives until mixture resembles coarse meal. Add buttermilk, stirring until moist.
5. Drop dough in 24 portions (about 1 teaspoon each) into stock mixture. Cover and cook 8 minutes. Gently stir in reserved shredded chicken and thyme. Simmer over very low heat for 6 minutes or until thoroughly heated and dumplings are done. Remove from heat, and ladle into shallow bowls. Sprinkle evenly with chopped parsley.