Freeze beef partially, until firm, about 1 hour. Slice across the grain into 1/2-inch slices.
Combine 1/3 cup soy sauce, green onions, sake, 1 tablespoon sesame oil, sugar, garlic, ginger, and black pepper in a large bowl. Add beef; toss to coat. Cover with plastic wrap and refrigerate until sauce flavors are absorbed, 1 to 3 hours.
Stir gochujang, 1 teaspoon soy sauce, and 1 teaspoon sesame oil together in a bowl.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler. Arrange marinated beef on a rimmed baking sheet.
Broil beef; stirring frequently, until glazed and dark brown, and nearly all marinade has evaporated, 15 to 20 minutes. Sprinkle with sesame seeds.
Arrange cooked beef, rice, and cucumber in serving bowls. Wrap portions of each in lettuce and dip into the gochujang sauce.