In a medium bowl whisk the marinade ingredients until the sugar is dissolved.
Plunge the bok choy into a bowl of water and shake to clean off the dirt nestled between the leaves. Wipe the mushrooms clean with a damp cloth or paper towel and then remove and discard the stems.
Place the vegetables in a large, resealable plastic bag and pour in the marinade. Press the air out of the bag and seal tightly. Turn the bag to distribute the marinade and let stand at room temperature for 1 to 2 hours, turning occasionally.
Prepare the grill for direct cooking over medium heat (350° to 450°F) and preheat the grill pan.
Remove the vegetables from the bag and reserve the marinade. When the grill pan is hot, place the bok choy and mushrooms onto the pan, spreading them out in a single layer. Grill over direct medium heat, with the lid closed as much as possible, until the bok choy is crisp-tender and the shiitakes are heated through, about 5 minutes, turning once or twice and basting occasionally with the reserved marinade. Serve hot.