Place chicken, broth, and onion in a large Dutch oven. Cut 1 celery stalk and 1 carrot in half crosswise; add to Dutch oven with thyme or parsley sprigs, salt, and pepper. Bring to a boil over high; reduce heat to low. Cover and simmer, undisturbed, until chicken is cooked through and easily pulls away from bone, about 20 minutes. Remove chicken; let cool slightly. Using 2 forks, pull chicken apart into bite-size pieces; discard skin and bones, herb sprigs, and cooked vegetables, reserving broth in Dutch oven.
Prepare Rolled Dumplings: Stir together all-purpose flour, baking powder, and kosher salt in a medium bowl. Using a fork, cut in butter until flour mixture is crumbly. Stir in buttermilk until dough comes together. Roll out dough to ½-inch thickness on a well-floured work surface; cut into 2- x 1-inch rectangles.
Return broth to a simmer over medium-low. Thinly slice remaining carrot and celery stalk, if desired; add to broth with pulled chicken. If desired, whisk together cream and flour in a small bowl until smooth; stir into broth. Gradually add dumplings to broth. Cover and simmer gently over low, until dumplings are cooked through, about 8 minutes. Remove from heat. Garnish with pepper and chopped thyme or parsley.
