In a large skillet, heat the oil until shimmering. Add the shiitake stems and cook over moderate heat, stirring a few times, until deeply browned, 5 minutes. Add the water and broth and bring to a boil over high heat. Boil until reduced to 1 cup, about 12 minutes. Strain the broth and return it to the saucepan. Boil over high heat until reduced to 1/4 cup, 7 minutes.