In the bowl, stir together the soy sauce, sesame oil, hoisin sauce, and cornstarch. Add the pork, carrots, and mushroom and stir until coated. Allow to stand for 10 minutes.
Pour half the oil in the wok and heat over medium-high heat. Add half the onions and stir-fry for 30 seconds. Add the bean sprouts and water and stir-fry for 2 minutes. Transfer to a medium bowl.
Add the remaining oil and warm. Add the remaining green onions and garlic, and stir-fry for 30 seconds. Add the pork mixture and stir-fry just until the meat is cooked, about 3 minutes. Add the bean sprouts mixture and stir. Serve about ½ cup of mixture in a lettuce leaf.