Whisk the soy sauce, brown sugar, rice vinegar, ginger and cornstarch in a small saucepan and simmer for 2 minutes, until glossy; cool half for the raw chicken and reserve half for serving.
Coat the chicken in the cooled marinade and refrigerate for 30 minutes.
Heat a grill or grill pan to medium-high and grill the drained chicken for 4 to 6 minutes per side, brushing with the chicken-contact marinade only during the first half of cooking.
Verify the chicken reaches 165°F, move it to a clean plate and rest for 5 minutes.
Grill the pineapple rings for 2 minutes per side and toast the buns for 1 minute.
Toss the cabbage with mayonnaise and assemble each bun with 1/2 cup cabbage slaw, 1 chicken cutlet, 1 pineapple ring and 1 tablespoon reserved teriyaki glaze.