As the shiitake is a pretty strong flavor that takes over other flavors pretty insidiously I haven't wanted to use it as a substitute for recipes that call only for button mushrooms.
Buttery garlicy sauteed mushroom & parsley toasts! spread on some fresh ricotta first and it will help diffuse the flavor if you want to appeal to a broader audience.
When used as aromatics to transform water into a stock or broth, vegetables and funghi are pretty much spent at the end of the process. Their flavor leaches out into the liquid. While the mushrooms you boiled might retain a faint, ghostly bit of flavor, they would provide little more than texture if re-used as an ingredient in a new dish. Recipes ask you to add fresh mushrooms--or more, pre-soaked dried mushrooms--to mushroom stock when preparing risotto or soup to build up layers of flavor and intensify the role of the featured ingredient in the finished dish. (Unless dried, shitake have rather mild, delicate taste if a meaty quality. Given cost, I'd recommend using only the woody stems of fresh shitake in broth and reserve the caps for other purposes.)
Eliz makes a good point that much of their flavor was likely given to the broth, and by now they likely have lost much of that plus their consistency. Still, waste not want not. Perhaps puree/process them to a paste/etc, then either use that as a thickener in leu of other blander items, or add some tasty things to the puree & have a simple spread/etc?