Your cat vomiting up whole pieces of food is a sign that he's eating too much too fast. I don't think what you're seeing is poo. It COULD be a hairball.

My cat Shortie has hairball problems. I couldn't figure out why one of my cats was pooping in the middle of my bedroom floor. I happened upon it one night while getting up to go to the bathroom A cold, wet hairball between your toes in the middle of the night is NOT fun!!

It turned out to be a hairball which looks EXATLY like a piece of poo. Next time it happens, and I know this is gross sounding, try to take a closer look (pick it up with a piece of paper towel). A hairball will not be solid like a piece of poo.

I hope this helps.