Step 1Heat the oven to 350 degrees. Grease 2 (8-by-2-inch) round cake pans, and line the bottoms with parchment. Grease the parchment, then lightly flour the pans.
Step 2In a medium bowl, whisk together the wheat and all-purpose flours, the salt, baking powder and soda, and cinnamon. Set aside.
Step 3In the bowl of a stand mixer fitted with the paddle attachment, beat the eggs. With the mixer running, beat in the sugars, mixing to combine thoroughly. Slowly beat in the vegetable oil.
Step 4Beat in the dry ingredients until fully combined, being careful not to overbeat. Fold in the carrot and pineapple by hand.
Step 5Divide the mixture evenly between the two pans and bake 40 to 50 minutes, rotating halfway through for even baking, until the cakes have risen and spring back when gently pressed in the center. A toothpick inserted should come out clean.
Step 6Remove and cool the cakes on a rack before unmolding from the pans and frosting.
Step 1Cream together the cream cheese and butter until light and fluffy with the paddle attachment. Beat in the powdered sugar in three stages. Scrape the bottom and sides of the mixing bowl several times to make sure all lumps are creamed. Do not overwhip the icing; it can break and get soupy.
Step 2Blend in the pineapple and vanilla completely. This makes a generous 4 cups of frosting.