1. In a 3- to 4-quart covered pan over high heat, bring 2 quarts water to a boil. Immerse bean threads in water and stir to separate; cover and remove from heat. Let stand until noodles are soft, 3 to 5 minutes. Drain well. With scissors, cut into 6-inch lengths.
2. Rinse mushrooms and place in a small bowl with enough hot water to cover; let stand until soft, about 15 minutes. Lift mushrooms from soaking liquid (reserve liquid for another use or discard) and squeeze dry. Cut stems off mushrooms and discard. Cut mushroom caps into 1/4-inch-wide strips.
3. Rinse beef and pat dry. Chop beef into 1/4-inch pieces. In a bowl, mix meat with 1 teaspoon sesame oil, 2 tablespoons soy sauce, sliced green onions, 1 teaspoon sugar, and the garlic. Cover and chill at least 30 minutes or up to 2 hours.
4. Meanwhile, trim root ends from green onions; cut into 2-inch lengths, including green tops.
5. In a 12-inch nonstick frying pan or a 5- to 6-quart nonstick pan over medium-high heat, stir 2 teaspoons sesame oil and the yellow onion until slightly softened but not limp, about 2 minutes. Pour into a large bowl. Return pan to medium-high heat, and add 1 more teaspoon sesame oil, the carrots, and the green onion lengths. Stir until green onions are slightly wilted, about 1 minute; add to bowl. Return pan to medium-high heat and add 2 more teaspoons sesame oil, the mushrooms, and the green beans; stir until mushrooms are lightly browned, about 2 minutes. Add to bowl.
6. Return pan to medium-high heat; add 1 tablespoon sesame oil and the beef mixture. Stir until lightly browned, 2 to 3 minutes; add to bowl. Return pan to heat and add drained noodles, 3 tablespoons soy sauce, 1 tablespoon sesame oil, 2 teaspoons sugar, and 1/4 teaspoon pepper; stir until noodles are hot, 1 to 2 minutes. Pour into bowl with vegetable-beef mixture. Mix well, adding salt and pepper to taste. Pour into a bowl; sprinkle with sesame seeds.