1To make the sauce, in a saucepan over medium heat, combine the soy sauce, mirin, sugar, ginger and garlic. Bring the mixture to a boil, then reduce the heat to low and simmer until the mixture becomes slightly syrupy, about 5 minutes. Let cool.
2Place the flank steak in a large zippered plastic bag. Pour half of the teriyaki sauce over the steak and refrigerate for 8 hours or overnight. Reserve the other half of the sauce.
3Prepare a charcoal grill for medium-high heat. Clean and oil the grill grate. While the grill is heating, remove the steak from the marinade and let come to room temperature.
4Arrange the steak on one side of the grill and the green onions over the other side of the grill. Cover the grill and cook, turning the steak occasionally, until it is nicely seared and cooked to medium-rare (about 115–120°F). Transfer the steak and green onions to a cutting board, tent with foil, and rest for 10 minutes.
5Thinly slice the steak against the grain, then transfer it to a warm platter along with the grilled green onions. Garnish with sesame seeds. Serve with the rice, passing the reserved teriyaki sauce on the side.