Cook the chicken. Heat one tablespoon of olive oil in a large skillet over medium heat. Add the chicken pieces, season with salt and pepper, and cook until golden brown and cooked through, about 4 to 5 minutes. Transfer to a plate. Cook the vegetables. Add the remaining tablespoon of olive oil to the skillet. Add the broccoli, bell pepper, and snap peas. Cook until crisp-tender, stirring often, about 4 to 5 minutes.
To make the peanut sauce, whisk together soy sauce, peanut butter, hoisin sauce, honey, sesame oil, and grated ginger in a small bowl. Whisk in 2 or 3 tablespoons of water, depending on the consistency you prefer, and set aside.
Make the peanut sauce. While the vegetables cook, whisk together the soy sauce, peanut butter, honey, sesame oil, grated ginger, and 2 to 3 tablespoons of water until smooth and creamy.
Bring everything together. Return the chicken to the skillet with the vegetables. Stir in the garlic and cook for about one minute until fragrant. Add the cashews, then pour the peanut sauce over everything. Toss well until everything is evenly coated. Bring the sauce to a gentle simmer and cook for another minute or two until slightly thickened.
Garnish and serve. Top with sliced green onions and sesame seeds, then serve over rice, quinoa, cauliflower rice, or noodles.