Sprinkle chicken thighs evenly with 1 teaspoon salt and 1/2 teaspoon pepper. Place chicken, skin side down, in a medium Dutch oven; cook over medium-high, undisturbed, until skin is browned and crisp, about 10 minutes. Transfer chicken to a plate, reserving drippings in Dutch oven.
Add onion, carrots, and celery to Dutch oven. Cook over medium-high, stirring often, until just softened, about 6 minutes. Add all-purpose flour and cook, stirring constantly, 30 seconds. Stir in 6 cups water, stock base, and thyme sprigs. Return chicken to Dutch oven; bring to a boil over medium-high. Reduce heat to medium-low and gently boil, stirring occasionally, until vegetables are tender and a thermometer inserted in thickest portion of chicken registers at least 165°F, about 15 minutes. Remove chicken from Dutch oven (keep Dutch oven over medium-low heat) and transfer to a cutting board; let cool 5 minutes. Shred chicken, discarding skin and bones. Remove thyme from Dutch oven; discard. Return shredded chicken to Dutch oven. Stir in remaining 1 teaspoon salt and 1/4 teaspoon pepper.
Stir together self-rising flour, buttermilk, and butter in a medium bowl until batter is just combined. Bring soup to a boil over medium-high. Drop batter by heaping tablespoonfuls into boiling soup. Cover and reduce heat to medium. Cook, undisturbed, until dumplings are cooked through, about 18 minutes. Garnish with additional pepper and thyme leaves.
