Heat a charcoal or gas grill to moderately high heat, keeping part of the grill cool for indirect grilling, and put the rack about 4 inches from the flame. Alternatively, heat a panini press or George Foreman grill according device instructions. Combine the soy sauce and mirin in a small saucepan over medium-low heat and cook until the mixture begins to bubble, 2 to 3 minutes. Remove the pan from the heat and add the ginger, garlic, and the scallions.
Brush the cabbage slices with the oil and sprinkle with salt and pepper. Put the cabbage on the cool part of the grill, if using, and close the grill cover. Cook, checking and turning occasionally, until you can pierce the leaves easily with a sharp knife, 40 to 45 minutes. When the cabbage is tender, brush it liberally with the teriyaki mixture and move it to the hotter part of the grill. Cook, turning once or twice and brushing with more of the sauce, until it's browned, 3 to 5 minutes.
Drizzle the cabbage with any remaining teriyaki sauce, and serve hot or warm with lemon wedges.
If cooking with a panini press or George Foreman grill, press the cabbage wedges for about 5 minutes, flip, and repeat. When softened, remove from press and dress with sauce mixture.