In a small bowl, whisk together soy sauce or tamari, garlic, ginger, and lime juice. In another small bowl, whisk together corn starch and water. Add to the soy sauce mixture and whisk to fully combine.
Heat sesame oil and olive oil in a wok or a large skillet over medium-high heat. Add onion, bell peppers and broccoli and cook, stirring often, for 4-5 minutes or until tender-crisp.
Add snow peas and cook for about a minute.
Add chopped, cooked chicken and stir to combine. Reduce heat to medium-low.
Pour sauce mixture over the vegetables and chicken and toss to coat until the sauce is thickened and the ingredients are coated. Add walnuts & toss to combine.