Melt the butter over medium heat, swirling occasionally, until it foams and turns golden with a nutty aroma. Remove from heat and pour into a large mixing bowl, allowing it to cool for about 5 minutes.
Grate the carrots using the medium holes of a box grater.
Preheat the oven to 350°F and grease and line four 6-inch cake pans (or see notes for other pan options).
To the cooled brown butter, whisk in the honey or maple syrup, applesauce, eggs, coconut milk, and vanilla extract until smooth and fully combined.
In a separate bowl, whisk together the flour, cinnamon, ginger, nutmeg, cardamom, baking powder, baking soda, and salt.
Add the dry ingredients to the wet ingredients and stir until just combined, being careful not to overmix. Fold in the grated carrots, chopped nuts, raisins, and shredded coconut if using.
Divide the batter evenly between the prepared pans and bake for 24–28 minutes, or until a toothpick inserted in the center comes out clean.
Allow the cakes to cool in the pans for about 10–15 minutes before transferring to a wire rack to cool completely.
To make the frosting, beat the cream cheese and butter together until smooth. Add the honey, vanilla, salt, and arrowroot starch, and continue beating until light, fluffy, and spreadable. Start with 2 Tablespoons of arrowroot and increase as needed to thicken the frosting to the desired consistency.
Place one cake layer on a serving plate and spread a layer of frosting on top. Repeat with the remaining layers. Apply a thin crumb coat over the entire cake. Refrigerate for 20–30 minutes to set before finishing with the remaining frosting.
Notes
Garnish with toasted nuts, a sprinkle of cinnamon, or a drizzle of honey if desired.
Serve at room temperature for the best flavor and texture.
You can also make this in two 9-inch round cake pans.