Wash the kale and remove any tough stems. Chop the leaves into bite-sized pieces and set aside.

Heat the sesame oil in a large skillet or wok. Add the mushrooms, ginger, and garlic, and stir-fry until mushrooms start to become tender. Add the kale and broth and cover immediately. Cook on medium heat, checking often to make sure that the water hasn't all evaporated, until the kale is tender, about 5-9 minutes. (Add more broth if it becomes dry.) When the kale is tender, remove the cover and add the water chestnuts. Cook uncovered for 2 minutes. Stir in the remaining ingredients, cook for one more minute, and serve.