Finely grate the cheese using the smallest holes on a box grater or grate it in a food processor.
Add the cheese, optional carrot, mayonnaise, softened butter, onion, pepper sauce, and mustard to a large bowl. Mix until well combined. Alternatively, combine everything in a food processor for a smoother paste.
Check the consistency of the cheese paste. Add a little more softened butter if needed until it is creamy and easy to spread.
If using food coloring, divide the cheese paste into separate bowls. Add your desired color to each portion and mix well.
Spread the cheese paste generously onto slices of bread. For layered sandwiches, alternate slices of bread with different colors of cheese paste.
Trim the crusts if desired, then cut the sandwiches into your preferred shapes and serve.