Pre-heat the oven to 350 degrees. Mist a 9” x 5” loaf pan with cooking spray and set aside.
In a large mixing bowl, add the egg, applesauce, sugar, and vanilla extract and stir together until well combined. Add the melted butter and stir together.
In a separate mixing bowl, add the flour, baking soda, baking powder, cinnamon, and salt and whisk until combined. Add the dry ingredients to the wet ingredients and stir together until just combined into a batter. Stir the shredded carrots into the batter until combined.
Pour the batter into the prepared loaf pan. Bake for 45-50 minutes until a toothpick inserted into the center comes out clean.
Transfer the bread in the loaf pan to a wire cooling rack and cool in the pan for 10 minutes. After the 10 minutes, invert the pan to release the bread onto the cooling rack and then flip it over so that it’s right side up. Allow the bread to rest directly on the rack until cool.
To make the glaze, combine the cream cheese and powdered sugar in a small mixing bowl and beat together with a hand mixer. Add the ¼ teaspoon vanilla extract and the tablespoon of milk and continue to beat together until smooth.
Once the bread is cool, drizzle the glaze over the top of the loaf and then slice the loaf into 12 (3/4” thick) slices.