Preparation

• Center an oven rack and preheat oven to 325 degrees.
• To make the caramel sauce, gently stir together the water and sugar in a heavy saucepan over medium-high heat, being careful not to splash the sides of the pot. Stop stirring and allow the sugar to boil until it is a rich amber color. To check the true color of the caramel (because it will appear much darker than it actually is), simply tilt the pot to see a thin layer of the liquid – resist the urge to stir or stick an implement into the caramel, as it may cause the caramel to crystalize. Remove the saucepan from the heat and slowly pour in ¾ cup of cream and place the saucepan back on the stove over medium heat, stirring until combined. Remove the pan from the heat and stir in the vanilla and salt. Reserve 1 cup of the caramel for your frosting; the rest will be used to assemble the cake. Place all of the caramel in the refrigerator to cool while the cake bakes.
• To make the cake, whisk together the eggs, egg yolks, ½ cup of the sour cream, and the vanilla in a small bowl; set aside. In the bowl of a stand mixer fitted with the paddle attachment, blend the flour, sugar, baking powder, baking soda, and salt for 1 minute at low speed. Add the butter and the remaining 1 cup of sour cream and blend on low speed until the batter comes together. This will take some coaxing – you will need to stop the mixer often to scrape the batter from the paddle and the bottom of the bowl. Once the mixture has come together, mix on medium-high speed for an additional 90 seconds. The batter will be thick. Add the egg mixture in thirds, mixing each third into the batter until just combined and scraping the bowl as necessary.
• Divide the batter equally among the three prepared pans (there will be approximately 1 pound 2 ounces per pan) and smooth the tops. Bake in the middle of the oven until the centers spring back when lightly touched and a toothpick inserted in the middle comes out just barely moist, 25 to 30 minutes. Cool the cakes in their pans on a wire rack for 30 minutes. Flip the cakes out and lay them on a wire rack, top side up, to cool to room temperature. Leave the parchment paper on until you assemble the cake.
• Because the frosting is at its best when fresh, make it just before assembling the cake. In the bowl of a stand mixed fitted with the paddle attachment, cream together the confectioners’ sugar and butter on high speed until the mixture is thick, fluffy, and light in color, about 2 minutes. Gradually add the reserved cooled cup of caramel and mix on medium speed until combined. Switch to the whisk attachment and, with the mixer running on low speed, drizzle in the ¼ cup of cream. Turn up the mixer to high speed for 1 to 2 minutes or until the frosting is fluffy.
• To assemble the cake, first give the caramel sauce a good stir. If any of the cake layers are domed, slice off the domes. (They are great to nibble on while you’re building the cake!) Remove the parchment paper circles. Cut each cake layer in half to yield six thin layers. Build this cake using the three bottom layers first, so that the three top layers constitute the top half of the cake. Place one of the bottom layers cut side up on a serving plate. Using a metal spatula, spread a very thin layer of the caramel (2 to 3 tablespoons) over the cake, then spread a thin layer (½ cup) of the frosting over the caramel. Don’t worry if some of the caramel blends into the frosting, as this just makes it all yummier. Repeat these steps with the two remaining bottom layers, followed by two top layers (all top side up), spreading each layer with caramel first, and then with frosting. Place the very top layer top side down on the cake. Apply a thin layer of frosting all over the cake to make a “crumb coat.” Place the cake in the refrigerator for 10 minutes to firm up. Take it out and frost with the remaining frosting.
• The cake will stay fresh for up to 4 days under a cake dome at room temperature – if it lasts that long!