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.