Heat vegetable oil in a saucepan over medium heat. Add red bell pepper, onion, mushrooms, garlic, ginger, and red pepper flakes; cook and stir until softened, about 5 minutes. Transfer to a plate.
Cook and stir chicken in the same saucepan until no longer pink in the center, about 5 minutes. Return red bell pepper mixture to the saucepan. Add chicken broth, peanut butter, and peanuts. Simmer over low heat, stirring occasionally, until sauce is smooth, about 5 minutes.
Bring a pot of salted water to a boil. Add green beans and egg noodles; cook until egg noodles are tender, about 6 minutes. Drain. Stir green beans and egg noodles into the chicken and sauce mixture.