Run cooked noodles under cold water, then drain and gently pat dry with paper towel so they are as dry as as possible. Toss with the 1 tsp olive oil. Set aside.
Mix sauce ingredients together in a small bowl. Set aside.
Spray a large skillet with cooking spray. Season chicken lightly with salt and pepper. Over medium-high heat, cook chicken until brown and cooked through, about 3-4 minutes per side.
Remove chicken from skillet and set aside. In the same skillet, sauté garlic and chili until just fragrant, about 30 seconds.
Add onion and bell pepper to skillet. Sauté until just soft, about 5 minutes.
Add carrots and green onion to skillet. Sauté another 2-3 minutes.
Push veggies to the sides of the skillet. Add noodles to the center and sauté until just slightly browned, about 2-3 minutes.
Toss noodles, veggies, and chicken together in the skillet. Add in the sauce and continue to sauté for another couple of minutes, until the sauce has absorbed into the noodles and all ingredients are evenly distributed.
Remove from heat, then mix in the basil. Serve immediately.