Preheat the oven to 325°F (163°C). Grease and line the cake pans with parchment paper circles cut to fit the pans.
Melt the shortening in a small saucepan over low heat, then set aside to cool. In the bowl of a stand mixer fitted with a whisk attachment, beat the eggs, egg white, sugar, and lemon juice on medium-high speed for 15 minutes. The mixture should become thick and voluminous. Always beat it for the whole 15 minutes, though.
While the egg mixture is beating, whisk together the flour, spices, and baking soda in a large bowl till blended. Add the shredded coconut, raisins, pineapple, pecans, carrots, and melted shortening, then toss to combine.
When the egg mixture is ready, gently fold it into the flour mixture till completely incorporated. Divide the batter evenly among the prepared pans and bake for 30 to 35 minutes or till the tops spring back when gently pressed. (Bake for 35 to 40 minutes if using a 9-inch/24-cm round cake pan.)
Remove from the oven and let cool in the pans for about 10 minutes. Loosen the edge of each cake with a sharp knife, then turn the cakes out onto a wire rack and let cool completely before assembling.
Once the cakes are have cooled, prepare the buttercream. Combine the egg whites, maple syrup, and cream of tartar in a large heatproof bowl. Place the bowl over a small or medium-sized saucepan with 2 inches (5 cm) of simmering water in it. It is important that the bottom of the bowl does not touch the water. Heat the mixture till it reaches 160°F (71°C) on a candy thermometer, whisking constantly so the eggs don’t curdle or seize.
Remove from the heat and transfer the mixture to the bowl of a stand mixer fitted with a whisk attachment, or use a metal bowl and a hand mixer. Start whipping on low speed, gradually increasing to high over the course of 30 seconds. Continue to beat the mixture on high speed till stiff and completely cool. This could take up to 8 to 10 minutes, or longer if using a hand mixer.
When the meringue is ready, switch to the paddle attachment on the stand mixer. Add the shortening bit by bit, beating on medium-high speed till thick and smooth. Beat in the vanilla.
If the buttercream becomes loose and liquid, your meringue may not have been sufficiently cool when you added the shortening. Chill the mixture slightly in the fridge, then resume beating. It will emulsify eventually. If it curdles, the shortening may have been too cold. No worries! Don’t fret or throw it out; just keep beating! Beating the mixture fixes pretty much everything.
To assemble the cake: Place the first cake layer on a plate or cake stand. Spread about a 1/4-inch (6-mm)-thick layer of buttercream evenly over the cake. Place the second cake layer on top of the buttercream, then spread more buttercream on top. Finally, place the last cake layer, then frost the top and sides of the cake with the remaining buttercream. You don’t need a super thick layer, but you want enough buttercream to press the coconut flakes into.
Scoop up handfuls of coconut flakes and gently press them into and all over the surface of the cake.
For best results, chill the cake, uncovered, for about 30 minutes to secure the coconut flakes. If needed, fill in gaps by placing a little frosting on the back of a coconut flake and securing it to the area.