In a bowl, combine the 10 ounces cubed chicken breast with 1 tablespoon water, 3/4 teaspoon soy sauce, and 1 teaspoon cornstarch. Stir to coat and set aside for 5 minutes.

Heat 2 teaspoons vegetable oil in a large skillet over medium-high heat. Add the marinated chicken and stir-fry 1–2 minutes, until the chicken is lightly browned and no longer pink inside. Transfer the chicken to a bowl.

In the same skillet, add the 1 medium diced onion and sauté about 2 minutes, until softened. Add the 1 3/4 cups bean sprouts and stir-fry 1 minute. Return the cooked chicken to the skillet and stir to combine. Remove the skillet from the heat and let the mixture cool for about 5 minutes (cool enough so it won’t cook the eggs when combined).

In a separate bowl, lightly whisk the 6 large eggs with 1/4 teaspoon sesame oil until just blended.

Gently fold the cooled chicken-and-vegetable mixture into the whisked eggs, mixing just until combined.

Heat 2 tablespoons vegetable oil in a nonstick skillet over medium heat. For each patty, drop about 1/2 cup of the egg mixture into the skillet and use a spatula to gently shape it into a round patty. Add more oil between batches if needed to prevent sticking.

Cook each patty about 2–3 minutes per side, until golden brown and cooked through. Transfer finished patties to a paper towel–lined plate to drain excess oil. Repeat with the remaining mixture.

While the patties cook or after they are done, make the gravy: in a small saucepan combine 1 cup low-sodium chicken broth, 1 tablespoon oyster sauce, 2 teaspoons soy sauce, and 1/4 teaspoon white pepper. Bring the mixture to a gentle simmer over medium heat.

In a small cup, stir together 2 tablespoons cornstarch and 2 tablespoons cold water to make a slurry. Slowly whisk the slurry into the simmering sauce and continue to whisk until the sauce thickens to a smooth gravy (about 1–2 minutes).

Arrange the fried egg patties on a serving plate, spoon the hot gravy over them, and garnish with chopped green onions and sesame seeds if desired. Serve immediately.