Place the cashews on a baking sheet in a single layer. Toast in the oven until fragrant, about 5 minutes. (They will crisp up as they cool.)
Place the chicken pieces in a large bowl. Sprinkle with salt and pepper and toss to coat evenly.
In a large nonstick skillet, heat 1 tablespoon of the vegetable oil over high heat until very hot. Add half of the chicken to the skillet and stir-fry (stir as you cook) until lightly browned but not cooked through, about 3 minutes. Transfer to a plate.
Add the remaining tablespoon vegetable oil to the skillet; then add remaining chicken, garlic and white parts of the scallions. Stir-fry until the chicken is lightly browned but not cooked through, about 3 minutes. Return the first batch of chicken to the pan. Turn the heat down to medium and add the rice vinegar; cook until evaporated, about 30 seconds.
Add the hoisin sauce, soy sauce and water; cook, tossing, until the chicken is cooked through, about 1 minute. Remove from the heat. Stir in the scallion greens, cashews and sesame oil. Serve immediately.