Heat a large skillet over medium heat and add 1 tablespoon olive oil.
Add the sugar snap peas, red and green bell peppers, shallots and minced garlic to the skillet; season with 1/4 teaspoon salt and 1/4 teaspoon pepper and stir to coat.
Add the chopped cashews and cook the vegetables, stirring occasionally, until just softened, about 5 to 6 minutes; transfer the vegetables and cashews to a bowl and set aside.
Season the chicken pieces with the remaining 1/4 teaspoon salt and 1/4 teaspoon pepper.
Add the remaining 1 tablespoon olive oil and the toasted sesame oil to the skillet and increase heat to medium-high.
Add the chicken and toss to coat, then cook until browned on all sides and cooked through, about 6 to 8 minutes.
Stir in the low-sodium soy sauce and cook for 1 minute more, then remove the skillet from the heat and add the chicken to the bowl with the vegetables.
In a large bowl combine the chopped butter lettuce, chopped mango pieces, the cooked vegetables and chicken, sliced green onion, chopped cilantro and any reserved cashews; toss gently to combine.
Arrange the sliced mango on top of the salad and sprinkle with the chili powder.
Serve immediately; no dressing is required but add one if desired.