1. Preheat grill to medium-high heat.
2. Combine first 3 ingredients in a small bowl. Cover and chill.
3. Dissolve 1/4 teaspoon sugar in 3 tablespoons vinegar in a medium bowl, stirring with a whisk. Add cucumber and red onion; toss to coat. Cover and chill.
4. Place tuna in a food processor; pulse until finely chopped. Combine chopped tuna, green onions, next 3 ingredients (through salt), 1 1/2 tablespoons vinegar, and 3/4 teaspoon sugar in a bowl. Divide mixture into 6 equal portions with moist hands, shaping each into a 1/2-inch-thick patty. Press thumb in center of patty, leaving a nickel-sized indentation.
5. Place patties on grill rack coated with cooking spray; grill 4 minutes. Turn patties over; grill 4 minutes or until desired degree of doneness. Remove from grill; cover and let stand 5 minutes.
6. Spread 1 tablespoon mayonnaise mixture evenly on cut sides of each bun. Top bottom halves of buns with patties. Spoon cucumber mixture evenly on top of patties, using a slotted spoon. Arrange cilantro evenly over cucumber mixture. Cover with bun tops. Serve immediately.