I would argue that most foods taste better the next day (excluding foods that expire the next day). I find that there’s something about sitting in the fridge overnight that allows a dish to develop its flavors.
But what excites me about leftovers is that they can be used to create something entirely new, with entirely different flavors. This excites the French, too, who treat leftovers as “les dessertes de la table.” They also call leftovers “les parties interieures de la bête,” but I tend to not eat or cook with animal innards (ie organs and shit).
The bones, however, are a different story. Soups are the best way to utilize leftover meat pieces and vegetables — and leftover turkey carcasses. Here’s a picture of my mom’s famous (in our family) turkey noodle soup. She boils the leftover turkey scraps and parts in a big pot with water and carrots, celery, potatoes, and herbs. All day. Then after the broth has developed a rich color and tastes of turkey and sweet vegetables, she adds cooked egg noodles, pieces of turkey meat, and salt and pepper.
Oh, and she removes the carcass.