How to Make It
Prepare the Cake Layers: Preheat oven to 350°F. Beat salted butter with a heavy-duty electric stand mixer on medium speed until creamy. Gradually add granulated sugar, beating until light and fluffy. Add eggs, 1 at a time, beating just until blended after each addition. Whisk together flour, salt, and baking soda in a medium bowl. Add flour mixture to butter mixture alternately with buttermilk, beginning and ending with flour mixture, beating on low speed until blended after each addition. Stir in vanilla extract.
Pour batter into 3 greased (with vegetable shortening) and floured 8-inch round cake pans. Bake in preheated oven until a wooden pick inserted in center of cakes comes out clean, 22 to 24 minutes. Cool in pans on wire racks 10 minutes. Transfer from pans to wire racks. Cool completely, about 1 hour.
Prepare the Caramel Frosting: Bring unsalted butter and both brown sugars to a boil in a 3 1/2-quart saucepan over medium, whisking constantly. Whisk in heavy cream, and return to a boil. Remove from heat. Pour mixture into the bowl of a heavy-duty electric stand mixer fitted with the whisk attachment. With mixer running on medium speed, gradually add powdered sugar, vanilla extract, and fine sea salt. Beat until thickened and spreadable, 8 to 10 minutes. (If it becomes too thick, beat in 1 to 2 teaspoons hot water.)
Assemble the cake: Spread frosting between layers and on top and sides of cake. Sprinkle top of cake with flaky sea salt.