Make the cake layers: Preheat oven to 350 degrees. Lightly coat two 8-inch cake pans with cooking spray. Line bottoms with parchment; spray parchment.
Whisk together dry ingredients in a large bowl. Add remaining ingredients, and mix until smooth, about 3 minutes. Divide batter between pans.
Bake cakes until they pull away from sides of pans, about 1 hour. Let cool in pans on wire racks. Cakes will keep, covered, for up to 1 day.
Make the frosting: Place sugar, water, corn syrup, and whites in a heatproof mixer bowl over a pot of simmering water. Cook, whisking occasionally, until sugar dissolves and mixture registers 160 degrees on a candy thermometer.
Attach bowl to mixer. Whisk hot sugar mixture on high speed until stiff, glossy peaks form, about 7 minutes. Whisk in vanilla and salt.
Assemble the cake: Place a cake layer on a plate. Spread 2 cups frosting on top. Place remaining cake layer on top of frosting. Frost top and sides with remaining frosting. Cake is best eaten within 1 hour of frosting.