Heat the oil in a Dutch oven on medium heat. Dry the chicken with paper towels and sprinkle it with salt and pepper. Brown one breast at a time, on both sides, for about six minutes per side. When each breast is done, remove it to a plate.
To the pot, add the onions, carrots, and celery. Cook them for about four minutes. Add the chicken stock and the chicken and bring it to a simmer. Cover and simmer until the chicken is no longer pink in the middle, about 25 minutes.
While the soup is simmering, prepare the dumplings Whisk together the buttermilk, egg, butter celery leaves, and shallots. Then, sift in the flour, cornmeal, baking powder, salt, and pepper. Stir together just until a dough forms. With wet hands, roll the dough into one inch balls and place them on a plate.
Remove the chicken from the pot. Discard the skin and bones, and shred the chicken. Add the chicken to the pot, and one at a time, drop in the dumplings. Cover and simmer the pot for 15 minutes. Add the thyme leaves and serve.