For the pimiento cheese:
4 ounces sharp or extra-sharp cheddar cheese
1/4 cup mayonnaise
1 (2-ounce) jar diced pimiento peppers, drained
1/2 teaspoon Tabasco sauce
1/2 teaspoon Worcestershire sauce
For the sandwiches:
8 slices sourdough sandwich bread
4 tablespoons mayonnaise or room-temperature butter
For the pimento cheese: Grate the cheese on the small holes of a box grater and place in a medium bowl. Add the remaining ingredients, stir to combine, and smash with a rubber spatula to form a paste.
For the sandwiches: Place the bread on a work surface and spread 1/2 tablespoon of mayo or butter on one side of each slice. Flip half of the bread slices over and divide the pimento cheese over those slices (1/4 cup per sandwich), spreading it into an even layer. Top the cheese with the remaining bread slices, unbuttered-side down, to form 4 sandwiches.
Heat a large nonstick frying pan over medium heat until hot, about 4 minutes. Place 2 of the sandwiches in the pan and cook until golden-brown, 4 to 5 minutes per side. Remove to a cutting board and repeat cooking the remaining sandwiches. Cut each sandwich in half and serve.