Make the cake: Heat oven to 350°. Grease and flour two 3-inch-deep 9-inch cake pans; set aside. Whisk together flour, baking powder, and salt in a medium bowl; set aside. In the bowl of a stand mixer fitted with a paddle, beat butter, 1 2⁄3 cups sugar, and vanilla on medium-high speed until pale and fluffy. Alternately add the flour mixture and milk in three batches until just combined, to make a batter. In a large bowl, whisk together cream of tartar and egg whites (reserve yolks for filling) until soft peaks form; slowly add remaining sugar and continue whisking until stiff peaks form. Add to cake batter and fold until combined. Divide batter between prepared cake pans and smooth tops; bake until golden brown and a toothpick inserted in the middle of the cakes comes out clean, about 40 minutes. Let cool for 30 minutes, unmold, and cool completely. Using a long, serrated knife, halve both cakes horizontally to create four layers in all; set aside.
Make the filling: Whisk together 1 cup sugar and reserved yolks (from cake recipe, above) in a 4-qt. saucepan; whisk in bourbon and butter, and heat over medium heat. Bring to a simmer, whisking constantly, and cook until mixture thickens to the consistency of loose pudding, about 2 minutes. Remove from heat; let cool completely. Stir in raisins, pecans, coconut, and vanilla; set aside.
Make the icing: Combine the remaining sugar, corn syrup, salt, and egg whites in the bowl of a stand mixer; place the bowl over a saucepan of simmering water so that the bottom of the bowl does not touch the water. Cook, whisking often, until the sugar dissolves and an instant-read thermometer inserted into the egg whites reads 140°. Place the bowl on the stand mixer fitted with a whisk, and whisk the mixture on medium-high until tripled in volume and stiff peaks form.
Assemble the cake: While the icing whips, place one cake layer on a cake stand and top with 1⁄3 filling; repeat with remaining cake layers and filling, leaving top layer uncovered. When icing is ready, spread it over the top and sides of the cake until the cake is evenly covered, creating swirls, if you like. Chill before serving.