Make the mayonnaise. This is best done a day in advance. You can also use commercial mayonnaise to which you will need to add some grated ginger, sesame oil, soy sauce and a little samba oelek. If you make fresh mayonnaise, you will have a lot leftover. If you doctor commercial mayonnaise, add the flavorings to 1/2 cup mayo, but use lesser amounts.
Wash and slice cucumbers thinly. Toss them with the sugar and vinegar and let sit to marinate for at least 20 minutes, up to an hour.
Dice the tuna finely - no large than 1/4 inch. (It is easier to do when the tuna is very cold - even partially frozen.)
Mix all the burger ingredients well and form into two large patties.
Cover and refrigerate for 15-20 minutes. Heat your grill - charcoal or gas - to high.
When the grill is hot, oil the grate very well, as the tuna will stick.
Place “burgers” on the grill and cook about 2 minutes per side, depending on how rare you like your tuna. When you flip them, move them to a portion of the grill that is oiled but has not yet been used. This will reduce sticking on the second side.
Place cooked burgers on half a bun, top with a generous dollop of the mayonnaise, and arrange some pickled cucumber slices on top.
Sprinkle with toasted sesame seeds.