Stir together 1 cup mayonnaise, diced pimiento, Worcestershire sauce, and onion. Stir in shredded sharp Cheddar cheese. (Store in an airtight container in refrigerator up to 1 week, if desired.) Spread 1/4 cup pimiento cheese mixture on 1 side of a white bread slice; top with another bread slice. Lightly spread both sides of sandwich with mayonnaise. Repeat with remaining pimiento cheese mixture for desired number of sandwiches. Cook, in batches, on a hot griddle or large nonstick skillet over medium heat 4 to 5 minutes on each side or until golden brown and cheese melts.