Sprinkle chicken with 3/4 teaspoon of the salt. Select SAUTÉ setting on a programmable pressure multicooker (such as Instant Pot). (Times, instructions, and settings may vary according to cooker brand or model.) Select HIGH temperature setting, and allow to preheat 2 to 3 minutes. Add oil and chicken to cooker, and cook, without stirring, until well browned, about 7 minutes. Remove chicken from cooker. Add carrot, onion, celery, and garlic to cooker, and cook, stirring occasionally, 5 minutes. Sprinkle all-purpose flour over mixture in cooker, and cook, stirring often, 1 minute. Add stock to cooker; bring to a boil, stirring and scraping bottom to loosen any browned bits. Return chicken to cooker, and add potato. Press CANCEL.

Cover cooker with lid, and lock in place. Turn steam release handle to SEALING position. Select MANUAL/PRESSURE COOK setting. Select HIGH pressure for 3 minutes. (It will take 10 to 15 minutes for cooker to come up to pressure before cooking begins.) Carefully turn steam release handle to VENTING position, and let steam fully escape (float valve will drop). (This will take 2 to 3 minutes.) Remove lid from cooker. Press CANCEL.

Stir together self-rising flour and remaining 1/4 teaspoon salt in a medium bowl. Add buttermilk, stirring to combine. Place dough on a lightly floured surface, and knead gently once or twice until dough comes together. Pat into 1/2-inch thickness, and cut into 8 equal pieces. Round edges of dumplings to make a uniform shape. Stir pepper into chicken mixture in cooker. Place dumplings on top of mixture in cooker. Select SAUTÉ and HIGH temperature setting, and, if necessary, allow to preheat 2 to 3 minutes. Cover cooker with lid, and lock in place. Cook until dumplings are tender, about 5 minutes. Carefully remove lid, and sprinkle with thyme.