Make the glaze: Combine all the glaze ingredients in a 1-quart saucepan and bring to a boil. Lower the heat and simmer until the glaze coats the back of a spoon, about 5 minutes. Strain through a sieve and reserve in a warm place until tuna burgers are cooked. (The glaze can be prepared up to 2 days ahead and stored, covered, in the refrigerator.)
Make the burgers: Grind the tuna in a meat grinder or chop with a large sharp knife to the texture of hamburger meat. (Do not use a food processor, which will shred the tuna rather than chop it.)
Transfer the ground tuna to a bowl and combine with the garlic, mustard, cayenne, salt, and pepper. Mix thoroughly. Divide the tuna into 4 equal portions. Using your hands, roll each part into a smooth ball and then flatten into a compact patty.
Heat the olive oil in a large skillet over medium-high heat and sear the tuna burgers until browned and medium rare, 3 to 4 minutes a side. Serve each burger on a buttered toasted bun and spread with a tablespoon of warm glaze. Garnish the burgers with equal amounts of pickled ginger slices.