In a bowl, combine the cubed chicken with 1 tablespoon water, 3/4 teaspoon soy sauce, and 1 teaspoon cornstarch; set aside for 5 minutes to marinate.
Heat 2 teaspoons vegetable oil in a large skillet over medium-high heat. Stir-fry the marinated chicken 1–2 minutes until lightly browned; transfer to a bowl.
In the same skillet, add the diced onion and sauté about 2 minutes until softened. Add the bean sprouts and stir-fry 1 more minute. Return the chicken to the skillet and mix; let cool slightly.
In a separate bowl, lightly whisk the 6 eggs with 1/4 teaspoon sesame oil. Fold in the cooled chicken and vegetable mixture gently until combined.
Heat 2 tablespoons vegetable oil in a nonstick skillet over medium heat. Drop about 1/2 cup of the egg mixture into the skillet and shape into a round patty with a spatula.
Cook each patty 2–3 minutes per side, until golden brown and cooked through. Repeat with remaining batter, adding more oil as needed. Drain finished patties on paper towels.
Make the gravy by combining 1 cup chicken broth, 1 tablespoon oyster sauce, 2 teaspoons soy sauce, and 1/4 teaspoon white pepper in a small saucepan and bring to a simmer.
Whisk together 2 tablespoons cornstarch and 2 tablespoons cold water to make a slurry. Slowly whisk the slurry into the simmering sauce and cook until thickened into a smooth gravy.
Arrange the fried patties on a serving plate, pour the hot gravy over them, and garnish with chopped green onions and sesame seeds if desired. Serve immediately.