Preheat the oven to 350°F (175°C). Grease and flour a 9x13 inch baking pan and set aside.

In a large bowl, whisk together the flour, baking soda, cinnamon, nutmeg, cloves, and salt until well combined.

In a separate bowl, whisk 3/4 cup of the applesauce, mashed banana, melted coconut oil, eggs, and 1 teaspoon of the vanilla bean paste until smooth.

Pour the wet mixture into the dry ingredients and stir just until no dry streaks of flour remain; do not overmix.

Fold in the grated carrots and chopped nuts, if using, until evenly distributed in the batter.

Transfer the batter to the prepared pan, spreading it into an even layer and smoothing the top with a spatula.

Bake for 30–35 minutes, or until a wooden skewer or toothpick inserted in the center comes out clean or with a few moist crumbs.

Let the cake cool in the pan for 10 minutes, then carefully invert it onto a wire rack and cool completely to room temperature.

For the frosting, beat the softened cream cheese in a medium bowl until smooth, then add the remaining 1/4 cup applesauce and remaining 1 teaspoon vanilla bean paste and beat until creamy and well combined.

Once the cake is completely cool, spread the cream cheese frosting evenly over the top. Slice into 12 pieces and serve.