I already do that and believe me I save every bit of meat that falls off of those bones. I usually use all the cooking liquid as the base for my soups and stews.
I often throw the carcasses in a zip lock bag in my freezer until I have time on the weekend to cook them down and make up my soups, stews, etc. I frequently use my large stock pot and cook down several carcasses at once and make up several pots of different soups, stews and filling for pot pies that I can either serve the family over the weekend or freeze so I can pull them out as needed to use for quick family meals during the week.
Almost every weekend I have my large stock pot in use cooking up soups, stews or large pots of beans so I can have them ready to go for family meals during the week. I also use the cooking liquid for making gravies, cooking rice, stuffing, and just about anything else I cook. I am probably one of the most frugal home cooks you have ever seen. I tend to use up everything possible. I even save the liquid from when I cook up corned beef and cabbage to use for soup stock. It makes really wonderful split pea, lentil or bean soups.
I freeze, preserve, or use up virtually everything before it has a chance to spoil and I have become a master at taking leftovers and turning them into entirely different meals, so no one feels like they are eating leftovers. I also have come up with lots of tricks for streching the amount of meat used in each meal, and even the men who are pretty much straight meat and potatoes people have not had reason to complain.
We are eating more comfort foods like macaroni and cheese and other pasta dishes, casseroles, rice and beans, and hearty soups and stews more often than we did previously, but we still have a roast of some kind about once a week or more and I use all the leftovers for other dishes throughout the week, suplementing with the meals I have premade in my freezer.
We also usually have a sandwich night on Friday’s since everyone seems to be heading in different directions and needs to eat at different times that night, and about once a week we have a breakfast for dinner night, since we have diverse schedules and would otherwise not have the opportunity to all share a breakfast style meal together each week.
I like to take the chicken/turkey carcass, cook it down, take all the meaty pieces and liquid and mix it all into a huge pan of stuffing, preferably homemade cornbread stuffing, with some chopped boiled eggs, liver and gizzards boiled and chopped with diced onion.....with some pan gravy, it makes a great meal all by itself!......
Another variation is to use RICE instead of the stuffing, it’s great, too!............
Brown rice is for smart humans. White rice is slave food. So is white bread...got some homemade whole wheat/rye/caraway bread baking right now. Eating out I’ll eat white rice, you have no choice 95% of the time. But at home it is brown rice and brown bread