In a large bowl, whisk together 2 tablespoons of water, 2 teaspoons of sesame oil, and cornstarch until dissolved. Add the chicken and toss to coat. Set aside to marinate.

In a small bowl, whisk together the 4 remaining tablespoons of water, the remaining 1 teaspoon of sesame oil, hoisin sauce, oyster sauce, and soy sauce until well combined. Set aside.

Heat 1 teaspoon of vegetable oil in a large skillet over medium heat. Pour in the beaten eggs and cook for 3 minutes. Flip the eggs and cook for another 3 minutes or until set. Slice the eggs into thin strips.

In the same skillet over medium-high heat, add the marinated chicken strips. Cook for 6 minutes on one side, then flip and cook for another 6 minutes until golden brown and juices run clear. Remove the chicken and set aside.

Heat the remaining teaspoon of vegetable oil in the skillet over medium-high heat. Add the shredded cabbage, shiitake mushrooms, garlic, and sherry. Cook while stirring regularly until the cabbage starts to wilt, about 2 minutes.

Add the cooked chicken, egg strips, and the sauce mixture. Stir until the sauce thickens, about 5 minutes. Remove from heat and mix in the green onions. Serve immediately.