Prepare a medium-hot fire in a charcoal grill with a cover, or preheat a gas grill to medium-high.

To make the salsa:

1. Combine the Maui onion, green onions, pickled ginger, cilantro, 2 tablespoons Asian sesame oil, soy sauce, and lime juice in a bowl and mix well.
2. Set aside to allow the flavors to blend.

To make the patties:

1. Combine the tuna, panko, egg, mustard, garlic, honey, tablespoons sesame oil, salt, pepper, and cayenne in a large bowl.
2. Handling the tuna as little as possible to avoid compacting it, mix well.
3. Divide the mixture into 6 equal portions and form the portions into patties to fit the buns.
4. When the grill is ready, brush the grill rack with vegetable oil.
5. Place the patties on the rack, cover, and cook, turning once, just until opaque throughout, about 4 minutes on each side.
6. During the last few minutes of cooking, place the buns, cut side down, on the outer edges of the rack to toast lightly.

To assemble the burgers:

1. Brush the cut sides of the buns with the teriyaki sauce.
2. On each roll bottom, place a patty and an equal portion of the salsa.
3. Add the bun tops and serve.