Trim the bases off the bok choy and discard. Trim the leaves from the stems and cut both crosswise into bite-size pieces, keeping the stems and leaves separate.
Heat the light sesame oil in the sauté pan over medium heat, then add the scallions, ginger, garlic, shiitake mushrooms, and a pinch of sea salt and sauté for 30 seconds. Add the water, tamari, and the bok choy stems and sauté for 2 minutes. Add the bok choy leaves, lime juice, toasted sesame oil, and a pinch of sea salt and sauté until the bok choy is just wilted, about 2 minutes. Taste and add another squeeze of lime if you like. Sprinkle with the toasted sesame seeds and serve immediately.
Toast sesame seeds in a 350° F/175° C oven until aromatic and slightly browned, about 5 to 7 minutes.