Whisk egg white in a large bowl. Whisk in cornstarch and rice vinegar. Add chicken and stir until completely coated. Allow to sit for 10 minutes, then remove chicken and pat dry with a paper towel.
Meanwhile, rinse rice under cool running water until water runs clear. Set aside to drain.
Turn on a multi-functional pressure cooker (such as Instant Pot) and select Saute function. Add peanut oil and chicken; saute for 2 to 3 minutes. Add mushrooms, green onions, and garlic; saute until fragrant and mushrooms have softened, 6 to 7 minutes. Turn off Saute function.
Add vegetable broth, rice, frozen peas and carrots, snow peas, water chestnuts, and broccoli. Stir in pepper. Select Rice function and close and lock the lid. Set timer for 15 minutes.
Release pressure using the natural-release method according to manufacturer's instructions, for 8 minutes. Release any remaining pressure carefully using the quick-release method according to manufacturer's instructions. Unlock and remove the lid.
Fluff rice with a rice spatula, add sesame oil, and turn on Saute function. Whisk together remaining egg yolk and 2 eggs in a small bowl and stir into rice mixture, working slowly and in a circular fashion until eggs are distributed throughout the rice and are fully cooked, 3 to 5 minutes. Turn off Saute function and fold in bean sprouts and soy sauce. Serve immediately.