Prepare all necessary ingredients. Set the oven to preheat at 350°F (175°C).
In a sizable mixing bowl, whisk together the eggs, milk, salt, and black pepper. Add the crumbled bread and stir until the bread absorbs the liquid and softens. Incorporate the ground beef, onion, Cheddar cheese, and shredded carrot into the bread mixture. Once combined, transfer the mixture into a 9x5-inch loaf pan, pressing down gently to shape.
In a separate bowl, blend the brown sugar, ketchup, and mustard until smooth. Evenly spread this glaze over the top of the meat mixture in the loaf pan.
Place the pan in the preheated oven and bake until the meat is fully cooked and no longer pink in the center, approximately 60 to 75 minutes. Verify doneness with an instant-read thermometer, which should read at least 160°F (70°C) in the center.
Allow the meatloaf to cool briefly before slicing and serving.