Whisk 2 tablespoons water, 2 teaspoons sesame oil, and cornstarch together in a large bowl. Add chicken, toss to coat, and set aside.

Combine remaining 4 tablespoons water and 1 teaspoon sesame oil in a separate bowl. Whisk in hoisin sauce, oyster sauce, and soy sauce until well combined; set sauce aside.

Heat 1 teaspoon oil in a skillet over medium heat. Add eggs and cook for 3 minutes; flip and cook until firm and set, about 3 minutes more. Transfer to a cutting board and slice into thin strips.

Add marinated chicken to the skillet and cook over medium-high heat for 6 minutes. Flip and cook until golden brown on the outside and the juices run clear, about 6 minutes more; transfer to a plate.

Heat remaining 1 teaspoon oil in the skillet over medium-high heat. Add cabbage, shiitake mushrooms, garlic, and sherry; cook until cabbage starts to wilt, about 2 minutes.