Prep the Beef: Place the flank steak (1 1/2 pounds) in the freezer for about 30 minutes. This firms it up, making it much easier to slice thinly. Slice the beef 1/8 to 1/4-inch thick against the grain, trimming away any large pieces of exterior fat while leaving the marbling for flavor.
Season the Beef: Transfer the sliced steak to a large bowl. Sprinkle with the table salt (1 teaspoon) and pepper (1/4 teaspoon), then toss to coat. Add the cornstarch (1 tablespoon) and toss again until every piece is lightly coated.
Mix the Sauce: In a small bowl, whisk together the soy sauce (1/2 cup), mirin (1/4 cup), sake (2 tablespoons), both sugars (1 tablespoon granulated sugar + 1 tablespoon packed light brown sugar), ginger (1 tablespoon), garlic (4 cloves), and black pepper (1/4 teaspoon) until the sugars dissolve.
Sear the Beef: Heat a large nonstick skillet or well-seasoned wok over medium-high heat. Add the oil (1 tablespoon). Once shimmering, add the beef in a single layer, working in batches if necessary so the pan doesn't become crowded. Let the beef cook undisturbed for about 1 minute until nicely browned, then flip and cook another 1 to 2 minutes, just until cooked through. Transfer to a plate.
Cook the Broccoli: Add the broccoli (4 cups) and water (2 tablespoons) to the same skillet. Cover and steam for 2 to 3 minutes until bright green and crisp-tender. Remove the lid and cook another 30 seconds so any remaining water evaporates.
Bring Everything Together: Return the beef and any accumulated juices to the skillet. Reduce the heat to medium and pour in the teriyaki sauce. Stir frequently as the sauce gently simmers for 1 to 3 minutes, until it becomes glossy and coats the beef and broccoli. Avoid overcooking, as the sauce will continue to thicken after it's removed from the heat.
Finish the Sauce: Remove the skillet from the heat. Stir in the butter until completely melted, then stir in the toasted sesame oil just until incorporated. The butter gives the sauce a silky finish while the sesame oil adds its signature nutty aroma.
Serve: Spoon the teriyaki beef and broccoli over hot cooked rice. Garnish generously with sliced green onions and toasted sesame seeds, then serve immediately with extra sauce spooned over the rice.