Prepare a medium fire in a charcoal grill with a cover, or preheat a gas grill to medium.

Mix the mirin, sake, and soy sauce in a small, heavy fire-proof saucepan. Place the pan on the grill rack, bring the mixture to a simmer, and cook until reduced to a slightly thicker consistency, 15 to 20 minutes. Set the glaze aside (it will thicken slightly as it cools).

To make the mayonnaise, mix all of the ingredients in a small bowl. Cover and refrigerate until serving.

Mix the sesame dressing and ponzu sauce in a small bowl. Place the spinach in a large bowl and toss lightly with the dressing. Cover and refrigerate until serving.

Mix the mirin, sake, and soy sauce in a small bowl. Stir in the miso and blend evenly. Spread the chuck out in a large, shallow pan. Distribute the miso mixture over the meat and sprinkle evenly with the green onion, ginger, garlic, pepper, and salt. Handling the meat as little as possible to avoid compacting it, mix well. Divide the mixture into 6 equal portions and, with cupped hands, toss 1 portion of the meat back and forth from hand to hand to form a loose ball. Pat the meat portion lightly on a flat surface to form a patty to fit the buns. Press the center of the patty down with your fingertips to create a slight indention in the center of the patty. Repeat with the remaining meat portions.

When the grill is ready, brush the grill rack with vegetable oil. Place the patties on the rack and baste each patty with the glaze. Cover and cook until browned on the bottom, 3 to 4 minutes, basting each patty again with the glaze. Turn the patties and brush the cooked side with the glaze. Cook, brushing frequently with the glaze, until done to preference, 4 to 5 minutes longer for medium. During the last few minutes of cooking, place the buns, cut side down, on the outer edges of the rack to toast lightly.

To assemble the burgers, place an equal portion of the spinach on each bun bottom and top with a patty. Spread the mayonnaise over the cut sides of the bun tops. Add the bun tops and serve.