In a medium bowl, combine soy sauce, hoisin sauce, 1/4 cup sesame oil, vinegar, Sriracha, green onion, and garlic. Slice chicken into 1/2-inch thick slices on diagonal and add to bowl. Cover and refrigerate for 20 to 30 minutes.
Remove chicken from marinade.
Heat 2 tablespoons vegetable oil in a wok over high heat. When oil is very hot, add half the chicken and stir-fry for 2 minutes. Remove chicken and set aside on a platter. Add 2 more tablespoons vegetable oil to wok and repeat with remaining chicken.
Add chicken, ginger, and chicken broth to wok. Push chicken to the side and then add noodles, cover, bring to a simmer, and cook until noodles are softened, about one minute. Remove lid and stir to combine. Add snow peas, carrots, broccoli, and crushed red pepper. Cook, stirring as needed, until everything has cooked through and liquid has been absorbed, 6 to 8 minutes longer.
Drizzle in remaining 2 tablespoons toasted sesame oil, if desired, and season with salt and soy sauce, if necessary. Garnish with green onions, edamame, and peanuts, and serve immediately.