In zip top plastic bag, mix together marinade ingredients. Add chicken and refrigerate.
Prepare teriyaki glaze by combining all of the ingredients in saucepan over medium heat. Bring to a slow simmer and cook, stirring occasionally, until glaze thickens.
Caramelize onions and peppers in oil and butter in skillet over medium heat.
Set aside some of the teriyaki glaze for serving. Brush remaining teriyaki glaze on chicken and broil about 6 inches from heat. Continue basting with teriyaki glaze until chicken is cooked through and no longer pink inside (about 10 to 15 minutes).
To serve, pile shredded lettuce on plate. Divide chicken and caramelized vegetables among 10 tortillas. Drizzle teriyaki glaze on top and cover with shredded cheese.