In a small bowl, whisk together the broth, sherry, soy sauce, cornstarch, and chili garlic puree until smooth and set aside.
Place a large nonstick pan or wok over medium high heat and heat until a drop of water sizzles. Add 2 teaspoons of the oil and swirl the pan to coat the entire bottom; then, add in the chicken. Thoroughly cook the chicken, about 4-5 minutes. Remove from pan and place on a plate; keep warm.
Place the remaining teaspoon of oil into the pan and add the onion and ginger. Cook for 30 seconds to blend flavors and add in the broccoli. Stir-fry until tender yet crisp, about 2 minutes. Add in the broth mixture from step 1 and the water. Cook, stirring constantly until mixture thickens and bubbles, about 1 minute. Add back in the chicken, add the cashews, and heat through, about 1 minute.