When you make full-fledged chicken broth, that is with meat (not just bones), you wind up with a good amount of meat that has little to no character. Yet, if you're like me, you cannot throw it away. It's food.
Usually I just make
chicken salad. The chicken just becomes filler, but if you add the right things the composed salad will have flavor.
Sometimes I re-fry the meat in a very hot pan with onions and other seasonings. The results can go well in a taco or stuffed in a pita or over a bed of rice or noodles (depending on the seasoning).
Today I made broth and had two full chickens worth of meat leftover. Some of it will invariably become chicken salad tomorrow. But, tonight I had a little fun.
I topped a piece of puff pastry that was threatened by freezer burn with a caramelized red onion, fresh thyme, and chicken. I gave it all a liberal hit of olive oil and into a 400 deg. oven. Fifteen minutes later and a squeeze of lemon and we had a very nice chicken and onion tart:

As I mentioned, the chicken itself doesn't retain much character, but the olive oil, thyme, and lemon combined with the blast of heat and the onions produced a nice result.
So, what do you do with the broth meat by-product?
Best,
Michael