Toast coconut flakes in a dry skillet over medium-low heat for 2 to 3 minutes, stirring constantly until lightly golden. Transfer to a plate and cool for 3 minutes.
Spread 1 tablespoon peanut butter over one face of each bread slice, taking it almost to the edges.
Arrange banana rounds in one even layer over 2 slices, dividing them equally. Scatter cooled toasted coconut over the banana.
Close with remaining bread, peanut-butter-side down, and press gently. Cut each sandwich in half and serve cold or pack promptly in a closed container, keeping chilled with the rest of lunch.