In a resealable plastic bag, combine the chicken breasts and 1/4 cup of teriyaki sauce. Seal the bag and marinate in the refrigerator for at least 30 minutes, or up to 2 hours for more flavor.
Preheat a grill or grill pan over medium-high heat. Remove the chicken from the marinade and discard the marinade. Grill the chicken for about 6-7 minutes per side, or until fully cooked (internal temperature should reach 165°F). Let the chicken rest for 5 minutes before slicing.
In a large bowl, combine the mixed salad greens, shredded carrots, cucumbers, cherry tomatoes, red bell pepper, and green onions.
Slice the grilled chicken and place it on top of the salad mixture.
Drizzle the remaining 1/4 cup of teriyaki sauce and sesame dressing over the salad. Toss gently to combine.
Top with sliced almonds and crispy chow mein noodles, if using. Serve immediately.