Preparation

Make the cake

• Position racks in the bottom and top thirds of the oven and heat the oven to 350°F.

  Butter three 9×2-inch round cake pans and line each with a parchment round. Butter the parchment.

  In a medium bowl, mix the flour, baking powder, and salt. In a 1-cup liquid measure, mix the coconut milk with the vanilla.

  In a stand mixer fitted with the paddle attachment, beat the butter and sugar on medium-high speed until light and fluffy, 3 to 5 minutes. Scrape down the bowl. Add the eggs one at a time, beating well after each addition.
   
  Add about one-third of the flour mixture and mix on low speed until incorporated. Add half of the coconut milk and mix until incorporated. Continue adding the flour mixture and coconut milk, alternating the two and ending with the flour. Add the sour cream and mix until incorporated. Pour the batter into a large bowl.

  In a clean mixer bowl and using the whisk attachment, beat the egg whites on high speed until soft peaks form, 2 to 3 minutes. Using a spatula, gently stir a large spoonful of the whites into the batter to loosen it, and then fold the remaining egg whites gently into the batter.

  Divide the cake batter evenly among the prepared cake pans. Level the batter with a spatula. Set two pans on the top rack and the third on the lower rack. Stagger the pans on the oven racks so that no pan is directly over another. Bake, swapping and rotating the pans’ positions after 15 minutes, until a toothpick inserted in the center of each cake comes out clean, 25 to 30 minutes total. Cool on racks for 10 minutes. Invert the cakes onto the racks, remove the parchment, and cool completely.

Make the filling

• In a medium bowl, whisk 1-1/2 cups of the cream and the egg yolks.

  Combine the sugar and flour in a medium saucepan. Add the cream mixture and cook, whisking, over medium heat until smooth, 2 minutes. Bring to a simmer and cook, whisking, until thickened to a pudding consistency, 8 to 10 minutes. Remove from the heat. Stir in the coconut, butter, vanilla, and salt and let cool to room temperature.

  With an electric hand mixer, whip the remaining 1/2 cup cream to soft peaks. With a spatula, gently fold the whipped cream into the filling.

Assemble the cake

• Put a cake layer on a flat serving platter or a cake stand lined with strips of waxed paper to keep it clean while icing. Top the layer with half of the filling, spreading it evenly with an offset spatula almost to the cake’s edge. Repeat with a second cake layer and the remaining filling. Top with the last cake layer.

Make the meringue

• Put the sugar and egg whites in the metal bowl of a stand mixer (make sure it’s clean) and set over a pot of simmering water. Whisk constantly until the sugar melts completely, 3 to 4 minutes. Rub a small amount between your fingers to make sure all of the sugar grains have melted.

  Transfer the bowl to the mixer, fitted with the whisk attachment, and whisk at low speed until the mixture becomes completely opaque and begins to thicken, about 4 minutes. Raise the speed to medium and beat until thickened to soft peaks that barely hold their shape and flop over when the beater is lifted, 5 to 7 minutes. Finally, raise the speed to high and beat until glossy and thickened to medium-firm peaks that stand up stiffly but curl slightly at the tip when the beater is lifted, about 4 minutes more.

  Using an offset spatula, apply the meringue thickly over the entire cake—don’t worry about spreading it smoothly or you’ll overwork the meringue (you may not need all of it). Then, repeatedly poke your fingertips into the meringue, pulling it into spikes all over the cake. Remove the waxed paper strips.

  Using a kitchen torch, brown the meringue by holding the torch 2 to 3 inches from the meringue and waving the flame over the cake until it’s browned all over.

  Make ahead: You can bake, cool, wrap, and store the cake layers at room temperature for up to 1 day or freeze for up to 1 month. You can refrigerate the assembled cake (without the meringue topping) for up to 4 hours before decorating it. Wait to make the meringue until you’re ready to finish the cake.