Melt the sugar in a heavy-bottomed saucepan over medium heat until deep amber (about 350 degrees F if using a thermometer), 8 to 10 minutes. Remove from heat, then carefully whisk in the butter, standing back as it bubbles. Slowly whisk in the cream, again standing back from the steam. Stir in the salt. Cool completely.
Preheat oven to 350 degrees F. Grease and flour three cake pans and line the bottoms with parchment rounds.
Whisk the cake flour, baking powder, and salt in a medium bowl.
In a stand mixer, beat the butter and sugar on medium speed for 3 minutes until light and fluffy.
Add eggs one at a time, mixing until just combined after each.
Mix in the vanilla extract and 1/3 cup of the cooled caramel sauce.
With the mixer on low, add the flour mixture in three additions, alternating with the buttermilk in two additions, starting and ending with flour.
Divide the batter among the three pans and bake for 26 to 30 minutes, until a toothpick comes out clean.
Cool in the pans for 10 minutes, then transfer to a wire rack to cool completely before frosting.
Beat the butter until creamy, add powdered sugar gradually, then mix in 1/3 cup of the remaining caramel sauce and salt until fluffy.
Frost between each layer, apply a thin crumb coat, refrigerate for 20 minutes, then apply the final coat of frosting and drizzle with any remaining caramel.
Slice and serve, or refrigerate until ready to serve.