Sounds like you may be having regurgitation of food, coming back from your stomach and into the esophagus. This is called gastroesophageal reflux disease (GERD). If this is the cause, it could be lessened with slower eating and smaller bites but you will probably need prescription medicine to get it under control.
I'm no doctor, but it sounds to me like GERD (Gastroesophical Reflux Disorder). The acid coming back up into your throat irritates the lining of the esophagus thus causing the cough reflex. Or at least that's how I understand it. I suffer from this every time that I eat when my GERD is acting up (but not all the time).
My son has suffered that for years. We finally went to a university hospital where they did all the typical GERD related test and as soon as they discovered he actually had GERD, it seemed like they concentrated on that as being the source of his cough, and stopped listening to what he was telling them, that the cough would start as soon as he ate food. It didn't have enough time to get to his stomach and then start to regurge. But anyway, I finally stumbled across the answer in our local newspaper's doctors column. It is called gustatory rhinitis, and it is triggered by the act of eating food, any food. Now if you look up the definiton it will describe a conditon where the persons nose starts to run during eating, and doesn't mention any cough. I figured there was no reason that this could not also be the cause of his cough, draining down the back of his throat rather than a runny nose. We had our doctor prescribe a nasal spray called Atrovent, which because it acts locally in the nasal passages, it doesn't have any real effects on the rest of the body. It can be used several times a day if necessary and it has made a terrific difference in the amount and forcefulness of his coughing. Ask you doctor to try you on the Atrovent nasal spray and see if it makes any difference. But don't be surprised if you pharamacy has to order it as mine did, because I don't think many people use Atrovent in the nasal spray form. Good Luck