Preheat the oven to 325 degrees F, mist three 6-inch diameter cake pans* with non-stick spray, and line with circles cut from parchment.
Place the butter and brown sugar in the bowl of an electric mixer, and whip on medium-high speed (scraping the bottom and sides of the bowl with a silicone spatula occasionally) until pale and very fluffy ( a FULL 5 to 8 minutes).
Stir in the all-purpose flour, cake flour, cinnamon, baking powder, baking soda, salt, nutmeg, and cloves on medium-low speed, until just barely combined (there should still be streaks of flour).
Bake until the cakes feel springy and a toothpick inserted in the thickest part comes out clean or with a few moist crumbs (about 42 to 48 minutes).
Place the butter, cream cheese, lemon juice, and vanilla in the bowl of an electric mixer fitted with a whisk attachment.
Whip on medium-low speed until smooth.
Scrape the bottom and sides of the bowl with a silicone spatula, then add about half the powdered sugar.
Stir in the powdered sugar until combined, then scrape the bottom and sides of the bowl again, and add the remaining powdered sugar and the salt.
Stir together until combined, then turn the mixer up to medium-high speed and whip until the frosting is fluffy and stiff. (Be careful not to over-whip the cream cheese frosting, or it could become runny. As soon as the frosting looks stiff, turn off the mixer and use the frosting immediately.)
*Can also be made in two 8-inch diameter cake pans, two 9-inch diameter cake pans, one 9x13 rectangular baking dish, or DOUBLE the recipe to make it in a 10- cup bundt pan. Adjust bake time as needed.
For the cupcake version of this layer cake, click here: Carrot Cake Cupcakes.
For more info on the frosting, click here: The Best Way to Make Cream Cheese Frosting.