In a large bowl, stir together both cheeses, mayonnaise, pimientos, banana peppers, Dijon, and hot sauce, if using, until well combined. Cover and refrigerate for up to 1 week.

Spread a thick layer of pimiento cheese mixture onto 5 bread slices. Cover with remaining bread slices to make 5 sandwiches.

Using a serrated bread knife, trim and discard crusts from sandwiches. Cut each sandwich into 4 rectangles. Serve immediately or cover with damp paper towels, place in a covered container, and refrigerate until ready to serve, up to 1 hour.