Preheat the oven to 180°C (350°F)
Grease three 6” or 7” cake pans with vegetable oil. I used pie tins because they were the only round pans I had on hand. Cut out circles from parchment paper to fit in the bottom of the pans. I did this by folding a piece of parchment paper into a long triangle and cutting it to fit the radius of the pan. Unfold, and you have a perfect fit for the bottom of the pan!
Make a buttermilk by mixing together the milk and vinegar. Let sit for five minutes.
Combine the dry ingredients in a large bowl (flour, baking powder, baking soda, cinnamon, nutmeg, cane sugar, and salt). Make a large well in the middle of the dry ingredients.
Make the batter: Into the well, add the milk and vinegar mixture, applesauce, oil, and vanilla. Start mixing together all of the liquid ingredients, then slowly incorporate the dry ingredients into the wet and mix it all together. Mix just until the batter is almost all smooth, with some lumps remaining. Add the shredded carrots and mix just until everything is incorporated. Do not overmix.
Divide the batter evenly among the cake pans. Smooth the tops and bake for 20 to 25 minutes. When a toothpick comes out clean, the cakes are ready.
Cool: let the cakes cool in their pans completely before frosting. Once cool, flip the pans onto a wire rack until assembly. Trim the edges evenly if needed.