Preheat oven to 350° F. Butter three 8-inch round cake pans, line with parchment rounds, and dust with flour, tapping out the excess.
In a stand mixer fitted with the paddle attachment, beat the butter, brown sugar and vanilla on medium-high speed until lighter in color and slightly increased in volume, about 5 minutes. Lower the speed to medium and add the eggs, one at a time, mixing until each is fully incorporated before adding the next.
Sift the flour, cocoa powder, baking soda and baking powder into a medium bowl. Add the salt to the dry ingredients after sifting, and whisk dry ingredients.
Alternate dry ingredients and buttermilk into creamed mixture, beginning and ending with dry ingredients. Mix until just incorporated, or finish by hand gently. Fold mayonnaise into batter with a whisk, until just blended.
Divide the batter evenly between the prepared pans. If possible, weigh the batter in each cake pan for 454 g each (excluding the pans--you will want to tare the scale each time.) This ensures even layers. Smooth with small offset palette knife, and bake for about 30 minutes, rotating once after 20 minutes. Cake is done when toothpick or skewer comes barely clean. Try not to over-bake. I tend to under-bake a few moments, so the skewer is a little bit gummy. This works well for a moist chocolate cake (not vanilla).
Let pans cool on wire rack for 20 minutes, then invert cakes onto racks, gently, peeling away parchment rounds. Let cool completely.
For the Buttercream:
Wipe the bowl of an electric mixer with paper towel and lemon juice, to remove any trace of grease. Add egg whites and sugar, and simmer over a pot of water (not boiling), whisking constantly but gently, until temperature reaches 160°F, or if you don't have a candy thermometer, until the sugar has completely dissolved and the egg whites are hot.
Place bowl back on mixer and fit with whisk attachment. Whip until the meringue is thick, glossy, and the bottom of the bowl feels neutral to the touch (this can take up to 10 minutes, or longer). Switch over to paddle attachment and, with mixer on low speed, add softened butter in chunks until incorporated, and mix until it has reached a silky smooth texture (if curdles, keep mixing and it will come back to smooth).
Add vanilla and salt, continuing to beat on low speed until well combined.
Assembly of the Inside-Out Neapolitan Layer Cake
Divide buttercream into 3, and flavor 1/3 chocolate and another 1/3 strawberry. Leave remaining buttercream vanilla. For chocolate, add melted and cooled bittersweet chocolate and blend until combined. For strawberry, add a few drops of Strawberry Flavor Oil (minimal) or 1-2 tablespoons of strawberry puree. Add a few drops of pink gel food coloring to strawberry buttercream
Trim any doming on your cake layers with a very sharp, serrated bread knife. Wrap layers in plastic wrap and chill for 15 minutes in freezer.
Spread a 1" circle of icing onto an 8" round thin cake board using an offset palette knife and place the first cake layer on top, face-up.
Using a small offset palette knife, spread 1 cup of chocolate buttercream on top, leaving about a 1" gap from edges. Place second cake layer on top, face up, and repeat, using Vanilla Buttercream.
Place final layer on top, face-down and repeat using Strawberry Buttercream.
Fill 3 piping bags with remaining buttercreams fitted with Wilton 8B decorative tip, or the tip of your choice, decorate edges, filling in gaps. Top with dark chocolate sprinkles.
*Keep buttercream in airtight container in refrigerator for up to one week, leaving out at room temperature when needed, rewhipping in mixer for 5 minutes.
**Can freeze buttercream for up to 6-8 weeks. To thaw, place on counter overnight, and rewhip for 5 minutes with paddle attachment in an electric mixer.