Preheat oven to 350°F (180°C). Spray 3 (9-inch) round cake pans with baking spray with flour.

In the bowl of a stand mixer fitted with the paddle attachment, beat butter, sugar, and oil at medium speed until fluffy, 3 to 4 minutes, stopping to scrape bottom and sides of bowl. Add eggs, one at a time, beating well after each addition. Beat in vanilla.

In a large bowl, whisk together flour, baking powder, and salt. With mixer on low speed, gradually add flour mixture to butter mixture alternately with buttermilk, beginning and ending with flour mixture, beating just until combined after each addition. Divide batter among prepared pans (about 600 grams each), smoothing tops with a small offset spatula. Tap pans on a kitchen towel-lined counter several times to settle batter and release air bubbles; smooth tops again, if necessary.

Bake until a wooden pick inserted in center comes out clean, 25 to 30 minutes. Let cool in pans for 10 minutes. Remove from pans, and let cool completely on wire racks.

Level cooled cake layers, if desired. Place 1 cake layer on a serving plate. Working quickly, pour and spread 1 cup (300 grams) Cooked Caramel Icing onto cake layer. Top with second cake layer; pour and spread 1 cup (300 grams) icing onto cake layer. Top with remaining cake layer. Pour remaining icing on top of cake, letting it run down sides, gently smoothing with an offset spatula. Let stand until icing is cool and set. Store in an airtight container for up to 3 days.