Cut fish into 1 inch cubes. Place in a single layer on a rimmed baking sheet, cover with plastic wrap, and transfer to freezer. Chill until very cold (it should feel firm but still give when pressed with your finger), about 15 minutes.

Meanwhile, sterilize all grinding parts (the stand mixer's grinding attachment, including the dies, or the bowl and blade of the food processor), rinse, dry, and place in the freezer until very cold, at least 15 minutes. Assemble grinder attachment with chilled coarse die. Set a bowl in a large bowl of ice, arrange beneath the attachment to catch ground tuna and keep it cold.

Combine tuna, scallions, ginger, and garlic in a large bowl. Grind mixture in batches, with mixer on medium speed, and plunge gently to press through grinder. Add mayonnaise, grapeseed or olive oil, and toasted sesame oil to ground tuna mixture. Mix gently just to combine.

Form into four 1/2 in thick patties. Season with salt and freshly ground black pepper as desired. Grill or pan fry burgers over medium heat until just cooked through, 3-4 minutes per side.

Split and toast the rolls. Assemble with lettuce leaves on the bottom of the roll, top that with a tuna burger, then some tartar sauce, then sliced tomato. Top with the other half of the roll, enjoy.