If using dried seaweed, place the seaweed in a small bowl, add enough water to cover, and set aside to soften for 15 minutes. Drain and coarsely chop before using.
Prepare the grill for direct cooking over high heat (450° to 550°F).
In a medium bowl mix the onion, scallion, soy sauce, sesame oil, ginger, and chile. Add the seaweed.
Brush the tuna with oil and lightly season with salt and pepper, patting the seasoning into the fish. Brush the cooking grates clean. Grill the tuna over direct high heat, with the grill lid open, just until seared on both sides but still rare inside, about 2 minutes, turning once.
Transfer the tuna to a cutting board and cut into ½- to ¾-inch cubes. Add to the bowl containing the onion mixture and toss to combine. Divide equally into individual bowls. Sprinkle with the sesame seeds. Serve warm with lemon wedges, if desired.