Preheat your oven to 350°F (175°C). Grease and flour a baking pan.
In a large mixing bowl, whisk together the eggs, honey, applesauce, and vanilla extract.
Add the grated carrots to the wet ingredients and mix well.
In another bowl, combine the whole wheat flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Pour the batter into the prepared baking pan and spread it evenly.
Bake for 30 minutes, or until a toothpick inserted into the center comes out clean.
Let the cake cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.