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To make the cake:
In a medium bowl, beat eggs with a whisk for 1 minute. Whisk in vanilla, applesauce, pineapple, carrots, and sweetener. Set aside.
Preheat oven to 350ºF. Prepare two 8-inch round cake pans by spraying with non-stick cooking spray, and dusting with flour.
In a small bowl, whisk together flour, baking powder, baking soda, spices, and salt. Add the dry ingredients to the wet ingredients and stir together just until moistened. Fold in nuts, if using.
Divide batter between the two prepared cake pans, smoothing out the batter on top. Bake for 30 minutes, or until a toothpick comes out clean in the middle.
Remove, let cakes cool slightly, then remove them from the pans onto a wire rack or parchment paper to finish cooling before frosting.
To make the frosting:
In a medium bowl, beat cream cheese, yogurt, and vanilla until completely smooth. Add in half of the sweetener and beat until incorporated. Add the rest of the sweetener, and beat. Add a little more sweetener if needed for a spreadable consistency, and beat again.
Split the frosting between the two cakes carefully spread all around the cake. Decorate the middle of the top cake with walnuts, pecans, or carrot ribbons.