In a small bowl, combine smoked paprika and brown sugar. Place pork tenderloin on a foil lined baking tray. Rub paprika mixture all over pork.
Roast pork about 15 minutes until it reaches an internal temperature of 145°F. Allow at least a 3-minute rest. Remove from oven and cool completely.
In a medium bowl, combine cheddar cheese, mayonnaise, pimentos, Worcestershire sauce, and cayenne if desired. Mix until ingredients are evenly distributed.
Thinly slice cooked tenderloin and season with salt and pepper.
Toast and butter brioche slices and evenly distribute sliced pork. Top each with a quarter of the pimento cheese spread. Finish with bibb lettuce and second piece of bread.
NOTE: Store bought pimento can be used instead of making scratch version. Any appropriately flavored pre-marinated pork tenderloin could also work in place of the smoked paprika seasoning.