To a large mixing bowl add the melted butter, Greek yogurt, applesauce, dark brown sugar, eggs, and vanilla extract. Stir all of the ingredients together until well combined.
Next, add the flour, baking soda, cinnamon, ground nutmeg, ground ginger, and salt to the wet mixture. Stir until everything is well combined.
Once well combined, fold the shredded carrots and walnuts into the batter.
Transfer the carrot cake bread batter to a 9×5 pan that has been lined with parchment paper.
Bake the bread at 350 degrees for 60-70 minutes or until a toothpick inserted into the center of the bread comes out mostly clean with no raw batter. Cover the bread loosely with foil, halfway through, to prevent it from becoming too brown if necessary. Let the bread cool in the pan before transferring it to a wire rack.
While the bread is cooling make the cream cheese frosting. Add the room temperature cream cheese, powdered sugar, cinnamon, and vanilla extract to a mixing bowl. Use a hand mixer to whip until well combined and smooth. Next, add a tablespoon of orange juice and continue to whip until fluffy. You can add another tablespoon of juice if you want a thinner consistency.
Once the bread has cooled completely, spread the cream cheese frosting on top of the bread. Slice, serve, and enjoy!