Place steak in a bowl; top with minced onion, garlic, and 1 tablespoon soy sauce. Stir to coat.
Bring rice and 4 cups water to a boil in a saucepan over high heat. Reduce heat to medium-low, cover, and simmer until the rice is tender, and the liquid has been absorbed, 20 to 25 minutes.
Heat vegetable oil in a skillet over medium heat; cook and stir bamboo shoots, mushrooms, 1/4 cup green onion, green bell pepper, red pepper flakes, and 3 tablespoons soy sauce in the hot oil until green bell pepper is slightly tender, 4 to 5 minutes. Add steak, broccoli, and sugar snap peas; cook until beef is browned and no longer pink in the center, about 10 minutes.
Whisk cornstarch and 2 tablespoons water together in a bowl until cornstarch is dissolved. Heat oyster sauce in a saucepan over medium-low heat; slowly stir in cornstarch mixture until sauce is thickened, 3 to 5 minutes. Stir oyster sauce mixture into steak mixture; sprinkle with 1/4 cup green onions. Cook steak mixture until sauce is thickened, 7 to 8 minutes. Serve over rice.