Preheat oven to 350 degrees F (175 degrees C). Grease 1 muffin tin.
Whisk chickpea flour, baking soda, cinnamon, salt, nutmeg, and ginger together in a bowl. Make a well in the center; add applesauce, molasses, egg, butter, and vanilla extract. Stir until just combined. Fold carrots into the batter.
Spoon batter into the muffin tin.
Bake in the preheated oven until a toothpick inserted in the center comes out slightly wet, 18 to 20 minutes. Transfer to a wire rack to cool.