After making the soup several times and enjoying the super delicious broth, I started to realize that I could do something similar when making broth from leftover chicken bones.
The key, I think, is having some raw chicken to start out with. So, when I spatchcock a chicken, I always save the backbone (and any extra skin/fat I trim off the chicken).
Once I have the lovely browned bits, I take out the bones and such, and add about a cup of chopped onion. I saute that in the chicken fat and browned bits until the onions are fairly soft.
Then, and only then do I add water and the chicken bones. I also sometimes add in thyme and/or bay leaves, depending on what I plan to use the broth for.