In a small bowl, whisk together the mayonnaise, teriyaki sauce, honey and cayenne.
PREPARE THE BURGERS:
On a grill pan (or outdoor grill), quickly grill the pineapple and bell pepper rings until they have great marks on both sides. Remove and set aside. Season the ground beef with salt and pepper and form into 4 patties. Cook them on the grill pan (or grill) for 4 to 5 minutes on the first side, then flip them over, add a splash of teriyaki on each one, and place a slice of provolone on each patty and let it melt while the burger cooks through- 4 minutes or so more, until no longer pink in the middle.
Butter the buns and grill until golden. Spoon a little bit of the sauce onto the cut side of the buns and allow it to slightly soak into the bread.
Place the patties on the bottom buns and top each one with two pineapple slices, the bell pepper ring and lettuce. Put the bun on top, mash it all together and serve with dishes of extra sauce for dipping.