Place chicken, onion, carrots, celery, broth, thyme, salt and pepper in a 6-quart slow cooker. Cover and cook on low for 3 to 4 hours, beginning temperature checks at 3 hours and removing chicken as it reaches 165°F.
Cover the cooked chicken on a clean plate. Turn the cooker to high, stir in evaporated milk and replace the lid for 15 minutes until the base is very hot.
Mix cornstarch with cold water and stir into the hot liquid. Cut each small biscuit into 4 pieces and add pieces individually, pressing their bottoms into the liquid without stacking them.
Cover and cook on high for 40 to 50 minutes without lifting the lid, until a dumpling cut through the center is fluffy with no raw dough.
Pull chicken into large pieces and gently fold it into the cooker with peas. Cover for 5 to 10 minutes until chicken reaches 165°F again, then divide the thick chicken and dumplings among 6 bowls.