Melt the butter in a medium saucepan over medium heat. Cook until the solids separate and brown to a dark golden color, 7 to 10 minutes. Remove it from the heat and let cool at room temperature until it reaches 70 degrees, about 30 minutes. Don’t chill it. It needs to remain in liquid form. Set aside.
Preheat the oven to 350°F. Adjust the rack to the center of the oven. Spray two 10x2 inch round cake pans with pan spray, line them with a piece of parchment paper, then coat the paper lightly with pan spray.
Sift together the flour, almond flour, baking powder, and baking soda into the bowl of a standing mixer fitted with a paddle attachment or use a large bowl and a hand mixer. Turn the machine on low and mix for 30 seconds. Set aside.
Combine the granulated sugar and cinnamon in a medium bowl and stir to blend. Add the powdered sugar and stir to blend, then add the brown sugar and stir again to blend. In another medium bowl, whisk the eggs lightly. Add the sugar mixture and the vanilla, and whisk together quickly to combine well.
Turn the mixer on low and stream the sugar mixture into the dry ingredients. Once lightly incorporated, increase the speed to medium and whip for 2 minutes.
Reduce the speed to low and slowly stream in the melted butter. Be sure to scrape in all the browned bits from the bottom of the pan. Mix for 30 seconds on medium speed, then turn the mixture to high speed and mix for 3 minutes more, scraping down the sides of the bowl well.
Using a balloon whisk, gently fold in the carrots, coconut, and pineapple. Carrot cake batter will keep in the refrigerator for up to 2 weeks.
Pour the batter in the prepared pans. Bake for 45 minutes or until a toothpick inserted in the center comes out clean. Let cool in the pans on a rack for 5 minutes, then invert onto the rack, remove the pans and parchment, and cool completely before frosting. Wrapped airtight, carrot cake will keep at room temperature for up to 2 days or in the freezer for up to 2 weeks.
When the cake is completely cool, top with the frosting and spread evenly to coat. The frosted cake can be stored for 1 day at room temperature or refrigerated for up to 3 days.
Using a standing mixer fitted with a paddle attachment or a hand mixer, cream the cream cheese and butter together on the lowest speed. Stop the machine and scrape down the sides after 2 minutes.
When the cream cheese mixture is completely smooth and lump free, about 4 minutes, slowly add the powdered sugar. When it is fully incorporated, about 2 minutes, add the lemon zest, lemon juice, rum, and vanilla. Mix to combine. Cream cheese frosting can be used right away or stored in the refrigerator for up to 3 days.