Slice tops off rolls about 3/4 inch from top. With fork, scrape out insides of bottoms of rolls, leaving 1/2 inch wall all around. Set rolls aside.
In medium bowl, stir together mustard and mayonnaise until well blended. Stir in carrots until evenly coated with mustard mixture. Spoon 1/3 cup of the carrot mixture over bottom and up sides of each roll to form a nest. Place rolls and tops, cut side up, on baking sheet. Break and slip an egg into each carrot nest. Sprinkle 1 tablespoon cheese over each egg. Bake in preheated 325 F oven until whites are completely set and yolks begin to thicken but are not hard, about 30 to 35 minutes.