Make the teriyaki sauce: In a small saucepan, combine the soy sauce, mirin, sake, sugar, grated ginger, and grated garlic. Bring to a gentle simmer over medium heat and cook, stirring occasionally, until syrupy and reduced to about 0.5 cup, 6–8 minutes; it should coat the back of a spoon. Remove from heat and keep warm.
Prepare the patties: Divide the ground beef into 4 equal portions (about 5 oz each). Gently shape into 4.5-inch patties with a shallow dimple in the center of each.
Season and sear: Just before cooking, season both sides of the patties evenly with the kosher salt and black pepper. Heat the neutral oil in a large skillet or griddle over medium-high heat. Sear the patties until well browned, 3–4 minutes per side for medium (135–145°F/57–63°C), adjusting time to your preferred doneness.
Glaze: In the final minute of cooking, spoon some teriyaki sauce over each patty and flip once or twice to lacquer both sides. Transfer the glazed patties to a plate and let rest 5 minutes; reserve remaining sauce for assembly.
Toast buns: Wipe the skillet if needed and place the split burger buns cut side down until golden, 1–2 minutes.
Assemble: Spread Japanese mayonnaise on the cut sides of the buns. On each bottom bun, mound some shredded iceberg lettuce, set a glazed patty on top, and spoon on a little more teriyaki sauce. Cap with the top bun and serve immediately.