1Preheat oven to 325°. Lightly coat cake pans with nonstick spray and line bottoms with parchment paper; spray again. Whisk flour, cocoa powder, baking powder, baking soda, and salt in a medium bowl. Using an electric mixer on high speed, beat eggs and sugar until pale and doubled in volume, about 4 minutes.
2Add mayonnaise and vanilla and beat until just combined. With mixer on low speed, add dry ingredients in 3 additions, alternating with 1⅓ cups lukewarm water in 2 additions, beginning and ending with dry ingredients. Scrape batter into pans, dividing evenly.
3Bake cakes, rotating halfway through, until a tester inserted into the centers comes out clean, 25-30 minutes. Transfer pans to wire racks and let cool 20 minutes before turning out onto racks. Let cool completely.
4Frosting: Put chocolate in a medium bowl. Bring sugar, corn syrup, and ¼ cup water to a boil in a small saucepan over medium-high heat, stirring to dissolve sugar. Boil, swirling pan occasionally and brushing down sides with a wet pastry brush, until mixture turns a deep amber color, 10-12 minutes.
5Remove from heat and whisk in cream (mixture will bubble vigorously). Return to medium heat and cook, stirring, until smooth, about 2 minutes. Pour caramel over chocolate and stir until mixture is smooth; let cool, stirring occasionally.
6Using an electric mixer on medium-high speed, beat chocolate mixture, gradually adding butter, until frosting is thickened and smooth, about 1 minute.
7Chill, stirring occasionally, until stiff enough to spread easily, 20-25 minutes. Place 1 cake layer on a cake stand or platter. Spread 1 cup frosting over to come just beyond edges. Sprinkle with ¼ tsp. salt and place second cake layer on top, pressing down gently. Repeat process with 1 cup frosting and ¼ tsp. salt.
8Place third layer on top, and spread remaining frosting over top and sides of cake. Just before serving, sprinkle remaining ¼ tsp. salt over top of cake; drizzle slices with oil, if desired.