Spread butter on one side of each slice of bread. In a large skillet over medium heat, place half the bread slices butter-side down. Top each slice of bread with 2 slices cheddar, 1 slice provolone, 2 pineapple rings, and 1/4 cup teriyaki sauce. Top with a second slice of bread, butter side up.
Cook until bottom bread slices are golden brown, 2 to 4 minutes. Flip and cook until golden brown and cheese is melted, 2 to 4 minutes more. Serve immediately.