Prepare the sauce: Pulse the carrot and ginger in a food processor until finely chopped. Add the vinegar, sesame oil, soy sauce and sugar and process until smooth. Drizzle in 2 tablespoons water and combine; season with salt and pepper. Transfer the sauce to a bowl and set aside.
Prepare the burgers: Chop the tuna into chunks. Wipe out the food processor and add the tuna; pulse a few times to break up the pieces. In a bowl, mix 2 tablespoons olive oil, the soy sauce, lime juice, cilantro and ginger; season with salt and pepper. Pour over the tuna and process until well blended. Form into 4 patties; brush each lightly with the remaining 1 teaspoon olive oil.
Preheat a grill or grill pan. Once hot, add the burgers and cook for 2 minutes on each side for rare, or to desired doneness.
Toss the sprouts in the carrot-ginger sauce. Place the burgers on buns and top with avocado and sprouts.