Once you have strained the broth from the chicken onions and garlic, you will have a nice bowl of cooked chicken bits and pieces leftover that are basically a blank palette flavor wise that you can cook with in many different recipes.
I hate to throw ANYTHING away that I think can be used, so what I do is let the chicken bits and pieces cool off in the bowl for 1/2 hour or so, then I go through it and pick out all the edible meaty chicken pieces and use them in other recipes.
Some of my favorite recipes that are well adapted to shredded and cooked chicken:
Chicken, Poblano and Goat Cheese Tamales
Chicken Pot Pie
Chicken and Leek Soup
Chicken and Cheese Quesadillas with Mojo de ajo
Chicken Noodle Soup
Chicken “carnitas” Tacos (fry up the chicken with onions in chicken fat or lard)
Chicken Tarragon Salad with cranberries for Sandwiches
Chinese Chicken Salad