Sprinkle both sides of the chicken with the salt and pepper.
Put the pot on the stove, turn the heat to medium-high, and add the oil. When the oil is hot, add the chicken thighs and cook until they are well browned on the bottom, about 7 minutes. Use the tongs or spatula to turn them over and cook for 5 minutes more. Move the chicken to the plate (this is a job for an adult) and when it is cool enough to handle, shred or chop it.
Add the onion, carrot, celery, and garlic to the pot and cook until the onion is golden, about 7 minutes.
Add the broth to the pot, along with the chicken and any juices from the plate. Lower the heat to low and cook for 15 minutes (it should just be lightly bubbling while it cooks), or until a food thermometer inserted into the thickest piece reads 165 degrees.
Add the rice and cook for 5 minutes. Serve right away, or cover and refrigerate up to 2 days.