While the chicken is marinating, cut the stir fry vegetables and whisk together the stir fry sauce in a small bowl.
To cook, heat a large non-stick sauté pan or wok over medium-high heat. Add the cooking oil to the pan and let it heat up. When oil is shimmering and hot, add in the chicken in an even layer. Let the chicken cook for 1 to 1 1/2 minutes without stirring or disturbing it. Flip the chicken pieces and let cook for another minute (it's okay if it isn't cooked through yet). Place chicken on a plate or in a bowl and set aside.
Add the bell pepper and onion to the pan and sauté for 2-3 minutes, stirring occasionally. Add in the garlic, ginger, scallions, chilies, and nuts, and cook for about a minute, stirring often to prevent the garlic and ginger from burning (it's okay if the vegetables and nuts get some color on them, though).
Lower heat to medium and add the chicken back to the pan. Pour in the stir fry sauce, giving it another stir before adding to pan to make sure all the cornstarch is mixed in. Stir everything together to combine and let it come to a heavy simmer. Reduce heat to medium-low and let simmer for another 3-5 minutes to let the chicken finish cooking. Remove pan from heat and season with more Anytime Sauce, soy sauce, sugar and/or sesame oil to taste, if needed.
Serve hot with cooked rice. Refrigerate cooled leftovers; keep for up to a few days in the refrigerator.