Preheat the oven to 325°F. Coat three 9-inch round cake pans with cooking spray and then line the bottom of each pan with a round of parchment paper.

Place the egg whites and cream of tartar in the bowl of a stand mixer fitted with the whisk attachment. Whisk on high speed until soft peaks form. With the mixer still on, slowly sprinkle in 3⁄4 cup (150 g) of the sugar and continue to whip until the whites become glossy. Turn the mixer off, scoop the whites out into a large bowl, and set aside.

Place the butter, shortening, the remaining 21⁄4 cups (450 g) sugar, the vanilla, baking soda, salt, in the mixer bowl (no need to clean it) and switch to the paddle attachment. Beat until light and fluffy. Add the egg yolks, one at a time, waiting for each yolk to incorporate before adding the next. Add in the mango puree and mix. Occasionally stop the mixer and scrape down the sides of the bowl. The batter should look thick and rich.

Add one-third of the flour and mix on medium-low speed until incorporated. Add half the cream and beat to incorporate. Repeat, alternating between the flour and the cream, ending with the flour. Remove the bowl from the mixer and scoop about 1 cup of the whipped egg whites into the cake batter. Gently fold it in by hand with a large spatula to lighten the batter. Once the initial egg whites are incorporated, scoop out more egg whites and repeat until all the egg whites are folded in.

Divide the batter among the prepared pans (about 665 g each) and bake until a toothpick inserted in the middle of a cake comes out clean, 30 to 35 minutes. Check the cakes after 20 minutes and rotate them if they are baking unevenly. Let cool in the pans until just warm to the touch, 15 to 20 minutes, and then un-mold onto wire racks to cool completely. The cake layers will rise impressively while baking and then deflate a bit as they cool.

Lower the oven temperature to 300°F. Coat two 9-inch round cake pans with cooking spray and then line the bottom of each pan with a round of parchment paper.

Bring the cream to a boil in a medium pot. While the cream is heating, place the eggs, egg yolks, sugar, passion fruit puree, and cornstarch in a large bowl set on a damp towel (to keep it from shifting). Stir together. Once the cream is boiling, pour it over the mixture in the bowl while constantly stirring the custard with a whisk.

Divide the custard between the prepared pans (about 700 g each). Bake until the center just barely moves and the edges of the custard have started to brown, 35 to 40 minutes. Immediately run a thin knife around the custard edge to help release it from the pan. Let the custard cool to room temperature in the pans, then cover and refrigerate for at least 2 hours or up to 24 hours.

Place the egg whites, cream of tartar, and salt in a clean bowl of the stand mixer fitted with the whisk attachment. Beat on high speed until soft peaks form. Slowly sprinkle in 1⁄2 cup (100 g) of the sugar and continue to whip until the egg whites look glossy. Turn off the mixer.

Peel and chop the kiwis then place in a medium pot along with the remaining 11⁄2 cups (300 g) sugar. Bring to a boil, stirring constantly. Remove from the heat and pour the kiwi sugar syrup through a fine-mesh strainer into another medium pot, catching all the solids. Place the kiwi solids in a food processor or blender and process into a smooth puree.

Heat the kiwi syrup over medium-high heat until it reaches 225°F (the early thread stage). Once it reaches that point, remove from the heat and turn the stand mixer with the whipped egg whites in it back on to high speed. Drizzle the kiwi sugar syrup carefully into the bowl, making sure it doesn’t hit the whisk attachment or it will send the scalding hot syrup all over the place. Continue to drizzle the syrup until it is fully incorporated, then add the lime juice. Continue to beat for 8 minutes, or until the outside of the bowl is cool to the touch.

Switch to the paddle attachment and reduce the mixer speed to medium-high. With the mixer running, add the butter 2 or 3 tablespoons at a time, beating until incorporated before each addition. If the frosting looks really loose and thin, the egg whites are probably still too warm. Try placing the bowl in the freezer for 10 minutes to cool it down, then continue. Once all the butter is incorporated, reduce the speed to low and add the kiwi puree. Mix until just incorporated.

Place one cake layer on a cake stand or serving platter and peel off the parchment paper if you haven’t already. Un-mold the custard onto the cake; you may need to coax it out with a butter knife. Peel off the parchment paper. Repeat this process with the second cake layer, the second custard layer, and then the final cake layer on top.

Frost the cake with the kiwi buttercream. Place kiwi half-moons around the edge for decoration. You can place some of the sliced kiwis on top of the cake if you like. Serve immediately.