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