Cut each breast in half, so you have 4 pieces in total. Pound the chicken to a uniform thickness, if needed.
In a Ziploc bag, combine pineapple juice, soy sauce, brown sugar, and mustard. Add the chicken breasts and marinate overnight in the fridge.
Heat a grill over medium high heat and coat with a teaspoon of canola oil. Cook chicken 4-6 minutes on each side or until cooked through. Remove from pan and top with slice of cheese. Cover loosely with aluminum foil.
Add pineapple rings to grill pan and cook for 1-2 minutes on each side. Remove and set aside.
Butter buns and place butter-side down in grill pan. Cook for 30 seconds or until buns are lightly toasted.
Spread mayo on buns, if desired, then top with chicken and pineapple ring. Serve immediately.