In a medium saucepan, combine rice, 2 cups water, and 1 teaspoon salt. Bring to a boil. Reduce heat to low, cover, and cook until rice is tender and water is absorbed, about 15 minutes. Remove from heat and keep covered until ready to serve.

Meanwhile, finely chop 4 teaspoons each of peeled ginger and garlic. Trim ends from green beans, then cut crosswise into 1-inch pieces.

Heat 1 tablespoon oil in a large skillet over medium-high. Add green beans and season with a pinch each of salt and pepper. Cook until green beans are crisp-tender and browned in spots, 3–4 minutes. Transfer to a plate.

Heat 2 tablespoons oil in the same skillet over medium-high. Add shaved steak and chopped ginger. Cook, breaking up into 2-inch pieces, until steak is browned all over and cooked through, 3–5 minutes.

Stir in chopped garlic; cook until fragrant, about 30 seconds. Add all of the stir-fry sauce and ¼ cup water , stirring to coat the steak in sauce, about 30 seconds.

Return green beans to skillet with shaved steak. Stir in half of the sesame seeds. Cook until green beans are warmed through, about 1 minute. Season to taste with salt and pepper. Fluff rice with a fork. Serve sesame ginger beef and green beans over rice. Garnish with remaining sesame seeds.