Pound the chicken breasts until they are ½ inch thick. Marinate in ½ cup of hoisin sauce for 30 minutes. Lightly grease the grates of an outdoor grill. Heat the grill to medium-high heat. Add the chicken breasts and cook for about 5-10 minutes per side or until fully cooked (the internal temperature has reached 165 degrees F.). Slice and set aside.
In a medium bowl whisk together all of the sauce ingredients until well combined. Set aside.
Cook the linguini according to package directions. Drain and set aside.
Heat the sesame oil in a large deep skillet or wok over medium heat. Add green onions, garlic, and ginger. Cook for about 1 minute. Add the bell pepper and cook for another minute. Add the linguini, bean sprouts, sliced chicken, and the sauce. Stir and cook until the sauce has thickened, about 2 minutes.