In a medium size bowl add the first 6 ingredients, if they look lumpy be sure to sift them first. Mix and set aside.
In a small bowl combine 1 tsp baking soda and 1 tbsp of ACV and mix until it stops fizzing. Set a side.
In a bowl of a stand mixer beat butter and low carb brown sugar until light and fluffy.
To the bowl of the mixer add eggs and mix until combined, it will look lumpy. Add the dry ingredients, ACV mixture, vanilla extract and half & half. Blend until combined. Add chopped pecans and shredded carrots, run the mixer on low just until it all comes together.
Evenly divide the batter between 3 pans. Bake for 25 minutes or until a toothpick comes out almost clean, a few crumbs is okay. Run a knife around the edge of the pan to loosen the cake. Allow the cake to fully cool before frosting it.
Instructions for the Cream Cheese Frosting
To a bowl of a stand mixer add cream cheese, pyure, vanilla extract and 3 tbsp of heavy cream. Beat on high until the cream cheese is smooth with no lumps. Add the rest of the heavy cream and beat until the frosting is light and airy.
Assemble the cake
Place 1 cake layer on either a plate or a cake stand, spread about 1/2 cup of frosting evenly over the cake, repeat for the other 2 layers. Frost the outside of the cake and decorate as desired.