Prep the chicken and vegetables. Slice the chicken into thin, even bite-size strips so it cooks quickly. Slice the bell peppers into thin strips, cut broccoli into small florets, slice the carrot into thin coins, and mince the garlic.
Make the sauce. In a small bowl, whisk together soy sauce or tamari, honey, lime juice, cornstarch, water, sesame oil, black pepper, grated ginger, and minced garlic. The sauce should look smooth with no dry cornstarch clumps.
Cook the chicken. Heat 1 tablespoon oil in a large skillet over medium-high heat. Add chicken in a single layer and cook for 5 to 6 minutes, turning occasionally, until golden on the outside and fully cooked through.
Cook the vegetables. Add remaining oil to the skillet. Toss in bell peppers, broccoli, snap peas, and carrots. Cook for 2 to 3 minutes until slightly softened but still crisp.
Add sauce. Pour the sauce into the skillet. Stir to coat the chicken and vegetables, then cook for 2 to 3 minutes until the sauce thickens and bubbles.
Serve. Garnish with green onions and sesame seeds. Serve over rice or noodles.