Mix beef with 1 tsp. sugar, the salt, pepper, and 1 tbsp. oil in a bowl. Let marinate at room temperature 2 hours.
Whisk together remaining 1 tbsp. sugar, the vinegar, mirin, soy sauces, and fish sauce until sugar has dissolved.
Divide meat into 2 equal portions. Heat a large wok or a large (not nonstick) frying pan over high heat until very hot but not smoking. The metal will have a matte look and a drop of water flicked onto its surface should evaporate on contact.
Add 3 tbsp. oil to wok and heat until shimmering but not smoking. Add half the beef in a single layer and sear until a brown crust forms underneath, about 30 seconds. Flip cubes and cook 30 seconds on second side.
Add half each of red and green onions and cook, stirring, about 30 seconds. Add half the soy sauce mixture and shake wok to coat beef. Add half each of garlic and butter and shake wok to distribute evenly. Transfer to a bowl and keep warm.
Wipe wok clean and return to high heat. Repeat steps 4 and 5 with remaining 3 tbsp. oil, beef, red and green onions, sauce mixture, garlic, and butter.
Arrange watercress on a platter; top with beef and vegetables. Serve with sauce.
*Dark soy sauce is thicker, darker, and less salty than regular soy; find it at Asian markets and well-stocked grocery stores.