Using a spiralizer (or mandolin with a julienne blade), shred zucchini into long, thin, spaghetti-like strips. Set aside.
In a small bowl, whisk together the soy sauce, vinegar, honey, chili paste, and cornstarch. Set aside.
Heat 1 tbsp oil in a large skillet over medium-high heat. Season chicken with salt and pepper, and sauté until browned and cooked through. Remove and set aside.
Add the remaining tsp of sesame oil to the pan, along with garlic and ginger, and cook just until fragrant, about 30 seconds. Add in bell pepper and broccoli, and cook for about 1-2 minutes.
Stir in soy sauce mixture and bring to a boil. Reduce heat to medium, and cook until sauce is thickened.
Toss in zucchini noodles, and cook for another minute or two, just enough to allow the noodles to mix with the sauce.
Remove from heat, stir in chicken, and toss well. Divide onto 4 plates, and top each with scallion, and 2 tbsp of cashews.