Drizzle your skillet (see note) with oil. Saute chicken until cooked through. Transfer to a plate and cover to keep warm.
Place ramen noodles in the bottom of the skillet. Sprinkle seasoning packets over the noodles, then add peppers and onions to the skillet. Pour 1 1/4 cups of water into the skillet. Cover and cook over medium-high heat for 5-8 minutes until noodles are soft and liquid is absorbed.
Whisk together remaining 1/4 cup water, hoisin sauce, soy sauce, rice vinegar, sesame oil, and garlic and add to the skillet along with the cooked chicken and the cashews. Stir to combine.
Allow to cook for 3-5 minutes until cashews start to become tender. Top with green onions, cracked black pepper (optional) and serve hot.