In a medium bowl, whisk together the teriyaki sauce, soy sauce, brown sugar, grated ginger, and minced garlic. Set aside about 1/4 cup of this mixture for glazing later.

Add the bite-sized chicken pieces to the remaining teriyaki sauce mixture. Toss well to ensure each piece is evenly coated. Let it marinate for at least 15 minutes at room temperature, or up to 30 minutes in the refrigerator.

Heat the vegetable oil in a large skillet or wok over medium-high heat. Once the oil is shimmering, add the marinated chicken in a single layer. Cook for 5-7 minutes, stirring occasionally, until the chicken is cooked through and no pink remains.

Pour the reserved teriyaki sauce mixture over the cooked chicken in the skillet. Continue to cook for another 2-3 minutes, stirring constantly, until the sauce thickens and beautifully coats the chicken.

Briefly warm the tortillas in a dry skillet, microwave, or directly over a low gas flame to make them pliable and easier to roll.

Lay out each warm tortilla. In the center of each tortilla, layer a generous portion of shredded lettuce, shredded carrots, sliced red bell pepper, and the teriyaki chicken.

Sprinkle with sliced green onions and any optional toppings like sesame seeds or a drizzle of sriracha. Fold in the sides of the tortilla, then tightly roll from the bottom up to create your delicious teriyaki chicken wraps.