combine the turkey and marinade ingredients in a large bowl; mix well, cover, and refrigerate for 30 minutes. soak the mushrooms in warm water to cover for 30 minutes;drain. cut off and discard the stems and coarsely chop the caps. add the mushrooms and the remaining stuffing ingredients to the turkey, mix well and set aside. combine the sauce ingredients in a small bowl; set aside.
cook the cabbage leaves in a large pot of boiling water until they are limp, about 2 minutes. drain, rinse under cold running water, and drain again. to stuff each leaf mound 1/8 of the stuffing at the stem end, foldthe bottom of the leaf over the stuffing, then fold over the right and left sides; roll the whole package over once to enclose the stuffing.
place a 10-inch frying panover high heat until hot. add the vegetable oil, swirling to coat the bottom of the pan. add the ginger;cook, stirring, until fragrant, about 10 seconds. remove the pan from the heat and arrange the cabbage rolls seam side down in a single layer in the pan. pour the sauce over the rolls. return the pan to medium-low heat; cover and simmer for 25min.
lift the rolls out with a slotted spoon and transfer them to a serving platter. add the cornstarch solution to the pan juices; cook, stirring, until the sauce boils and thickens. pour the sauce over the rolls and sprinkle with green onion. serve with hot brown rice.