Preheat oven to 325 degrees and spray or line two 8 inch round cake pans. Combine the flour, baking soda, baking powder, spices, and salt in one bowl. Whisk together the yogurt, sugar, apple sauce, vanilla, and coconut oil in another. Make a well in the center of the dry ingredients and pour the wet ingredients into it. Fold the dry ingredients into the wet ones until everything is combined and then fold in the grated carrots. Divide the dough between your two prepared pans and stick in the oven for 30 minutes (or until a toothpick is inserted into the center and comes out clean).