Position shredding disk in food processor bowl; shred Cheddar cheeses. Cut cheese loaf into cubes. Position knife blade in processor bowl; add half of cheeses. Process until smooth, stopping once to scrape down sides. Transfer mixture to a bowl.
Add remaining cheeses, sugar, salt, and pepper to processor bowl; process until smooth, stopping once to scrape down sides.
Stir into cheese mixture in bowl. Stir in pimiento and mayonnaise.
Note: This dish may be frozen.