Heat a wok with first part of cooking oil over medium-high heat. Add chicken and saute until chicken is cooked through, 4 to 6 minutes. Set chicken aside and return wok to heat.
Add second part of cooking oil and then broccoli. Saute until broccoli is cooked through (add a splash of water and cover with a lid if you’d like the broccoli more tender), 3 to 4 minutes.
Add bell peppers, garlic, and white parts of green onions and saute for 1 minute. Pour sauce over top and bring to a simmer.
Stir chicken, noodles, and cashews into stir-fry. Toss everything together.
Squeeze lime juice over lo mein and top with sesame seeds and green parts of green onions. Enjoy!