Combine the first 6 ingredients in a small bowl. Cover and refrigerate the ginger sauce until needed. Preheat your grill to medium-high. Spread the 2 Tbsp. sesame seeds on a plate. Brush the tuna with the vegetable oil and season with salt and pepper.
Lightly oil the bars of the grill. Grill the tuna on all sides for about 20 seconds per side, or seared on the outside but still rare in the middle. Set the tuna on a plate and brush with the sesame oil. Transfer the tuna to the plate with sesame seeds on it and turn to coat.
Slice the tuna, widthwise, into thin slices and arrange on a platter. Sprinkle with green onions and serve the ginger sauce alongside for dipping.
Note: To toast the sesame seeds, place them in a small skillet over medium heat. Cook, swirling the pan from time to time, until lightly toasted.