Whisk eggs, milk, salt, and ground black pepper in a large bowl. Add crumbled bread and stir until dissolved. Mix ground beef, onion, Cheddar cheese, and carrot into bread mixture; transfer mixture to a 9x5-inch loaf pan. Combine brown sugar, ketchup, and mustard in a small bowl; spread over meat mixture.
Bake in the preheated oven until no longer pink in the center, 1 to 1 1/4 hours. An instant-read thermometer inserted into the center should read at least 160 degrees F (70 degrees C).