Preheat the oven to 325 degrees F. Prepare three 8-inch or four 6-inch round cake pans by brushing them with vegetable oil and then sprinkling with flour, shaking off the excess. (Note: just a small thin layer of oil and the same goes for the flour)
In a medium sized bowl, whisk together the all-purpose flour, baking powder and salt. Set aside.
In the bowl of a stand mixer, fitted with the paddle attachment, beat the butter, oil, and sugar on high until fluffy and smooth, about 3 minutes. Scrape down the sides and beat again for 2 more minutes.
Add the eggs and the egg whites, one at a time to the bowl and beat on medium until well incorporated. Add the emulsion and beat on low.
Add the sour cream and beat on medium speed for a minute or so. It may look curdled, this is normal. Once you add the flour it will look the way you would expect it to.
Turn the mixer on low and add ⅓ of the flour mixture to the butter mixture and mix until fully incorporated. Then add half of the coconut milk and blend well, followed by another ⅓ of the flour mixture, the second half of the buttermilk and lastly, add the remaining flour mixture and mix until incorporated.
Evenly distribute the cake batter among the three 8-inch or four 6-inch pans and bake for 23 to 28 minutes, until toothpick comes out with just a few moist crumbs on it.
for the lime curd
In a medium sized sauce pan, whisk the sugar, lime zest, and lime juice.
In a small bowl, whisk the whole eggs, and egg yolks. Slowly add the egg mixture to the lime/sugar whisk constantly.
Let the mixture cook on medium heat for about 7-8 minutes.
Remove from the heat and whisk in the butter. It might take a few minutes. I throw my butter in the mixture and let it sit for a couple minutes and then whisk it after it melts. (If you are adding food coloring, this is the time to add it. Remember, a little goes a long way)
Strain through a fine-mesh sieve into a glass or plastic tupperware/bowl. With a rubber spatula make sure you push the lime mixture through the sieve.
Cover with plastic wrap, pressing it down onto the mixture to help prevent a skin forming. Let it sit out until it comes to room temperature and then refrigerate at least 4 hours to set.
For the Coconut buttercream
In the bowl of a stand mixer fitted with a paddle attachment, beat the butter until creamed for about 2 minutes. This will help soften the butter.
Turn the mixer off, and add the sifted powdered sugar, whipping cream, coconut emulsion, and salt.
Starting with mixer on low, mix until everything is combined.
Turn the mixer to medium-high speed and beat for an additional 3 to 5 minutes. The frosting will become lighter in color and texture.
To help remove the air bubbles, beat with a wooden spoon or a rubber spatula before spreading onto the cake.
For the lime buttercream
In a small bowl, combine the curd, and the coconut buttercream, mix until combined.
Assembly
Level the cakes if needed. Place a small dollop of frosting on a cake board and place one cake layer top side up in the middle of the cake board.
With a piping bag, pipe a small ring around the edges of the cake.
Place about ½ cup of the lime frositng on the cake. Frost to the edge of the cake, then apply a thin layer of the lime curd on top of the buttercream
Place another one of the cake layers top side down and repeat step two and three.
Add the last layer of cake, top side down.
Apply a thin layer (crumb coat) of the coconut buttercream around the cake, and freeze for 15 minutes.
Apply the rest of the coconut buttercream and serve at room temperature.