Old mature roosters make the best broth but old hens aren't far behind. Even young cockerels and pullets make good broth. There are all kinds of different broth recipes: oven, stovetop, or pressure cooker. I use a crock pot.
When I butcher I take the time to part them, cut them into serving pieces. That's how I cook them anyway. I save the back, neck, gizzard, heart, and feet for broth. Yeah, I know where those feet have been. I scald them, twist the claws and spurs out, and skin them. That gets the feet clean enough for me but many people just throw the feet away or feed them to a dog. I freeze all this in a zip-loc bag. When I cook the chicken parts I save the bones. These go into broth too.
To make broth I put a bay leaf and a dozen peppercorns in a large crock pot. Toss in a carrot and stick of celery, a rough cut onion and some garlic. I add whatever herbs I feel like, parsley, chives, oregano, basil, or thyme can work. Then I put in the chicken and bones and top off with water. I cook that overnight on low, which means 14 to 20 hours. I strain the big chunks out, de-fat it, and then strain it through cheesecloth. Lots of broth.
But I don't stop there. I then go through the big chunks and pick the meat out. Be careful there can be some small bones. It's surprising how much meat you can get, neck and back both. Some people find this meat mushy but I think it is great on chicken tacos, in a chicken salad, or in soups and stews. I like it on bread as a sandwich for lunch.
There is a twist in this. You can use the entire rooster or hen if you wish, instead of cooking the parts with another recipe. You probably want to use half a hen for each batch so you have room for water to make broth. For a big rooster you might want three batches. Some people take the breasts and legs out after 8 hours, take the meat off the bones, and put the bones back in to finish making broth. That way the meat is firmer but still well-cooked and tender.
There are so many different ways you can cook our old hens and roosters. Did anybody mention Coq au Vin, I remember seeing chicken and dumplings. With a good recipe I'd think sausage would be great. I think I said it in a different thread, but if somebody says you can't cook an old chicken they are just admitting they don't know how.