I've never had a chicken with a foreign body lodged in the throat. However, just recently I had a hen with a growth blocking the trachea and she was humanely euthanized. Her symptoms did not include gasping for air. She simply could not keep food or water down. She would eject food with a shake of her head. I could actually palpate the growth by massaging her neck.
Chickens who are gasping or appear to be choking are usually suffering from bacterial or viral respiratory disease, pneumonia, or collapse of the air sacs. You will also have birds with crop issues appear to yawn as they swallow air. I did have a chicken with an ear infection yawn repeatedly probably to release the pressure on the inner structures of the ear.
Chickens who bite off more than can swallow will usually dislodge the object by violently shaking their head back and forth in an attempt to break it down into smaller pieces. Usually another chicken will grab the food item and run with it as other chickens follow eagerly grabbing at the object. This is their way of tearing apart large food items.
Healthy chickens rarely choke on food. Gasping chickens need a good hands-on exam to evaluate what is truly going on.