Position a rack in the center of the oven and heat it to 175°C or 350°F. Grease three 8-inch by 2-inch round aluminum pans, line the bottoms with parchment, and lightly grease the paper. In a medium bowl, whisk the flour, baking powder, and cake salt for 30 seconds. Distributing the leavener now prevents irregular tunnels and isolated salty spots in the finished layers.

Using a stand mixer with the paddle, beat the 170 g softened butter, 60 g oil, and 300 g brown sugar on medium speed for 3 minutes, scraping twice, until lighter and creamy. Add the eggs one at a time, mixing until each disappears, then add the cake vanilla. On low speed, alternate the dry mixture in three additions with the buttermilk in two, stopping when only a few flour streaks remain. Fold by hand until smooth.

Divide the batter evenly among the prepared pans, approximately 455 g per pan, and smooth the surfaces with an offset spatula. Bake for 23 to 28 minutes, rotating the pans once if the oven heats unevenly. The cakes are done when the centers spring back, the edges begin pulling from the pans, and a tester shows moist crumbs but no wet batter. Cool in the pans for 15 minutes, then turn onto racks and cool completely.

Combine the granulated sugar and water in a clean, heavy-bottomed saucepan. Cook over medium heat without stirring, swirling only as needed, until the syrup reaches a deep amber color, about 8 to 12 minutes. Remove it from the heat and slowly whisk in the warm cream, protecting your hands from the vigorous steam. Whisk in the 56 g butter, caramel salt, and caramel vanilla. Transfer to a heatproof bowl and cool until thickened but pourable, about 21°C or 70°F.

Reduce the oven to 135°C or 275°F and line a large rimmed baking sheet with parchment. Heat the 15 g oil with three popcorn kernels in a large covered pot over medium heat. When those kernels pop, add the remaining kernels, cover, and shake periodically until popping slows to one pop every 2 to 3 seconds. Transfer the popcorn to a large heatproof bowl, carefully discarding every unpopped kernel before adding caramel.

In a small saucepan, combine the 75 g brown sugar, 43 g butter, corn syrup, and popcorn salt. Bring to a steady boil over medium heat and cook for 2 minutes while stirring. Remove from the heat and stir in the baking soda; the mixture will foam. Immediately pour it over the popcorn and fold with a heatproof spatula. Spread on the prepared sheet and bake for 35 to 40 minutes, stirring every 10 minutes, then cool completely until crisp.

Beat the 340 g softened butter with the clean paddle attachment on medium speed for 3 minutes, until pale and creamy. Mix in the sifted confectioners’ sugar in four additions on low speed. Add 130 g of the cooled salted caramel sauce, the 30 g cream, buttercream vanilla, and buttercream salt. Beat on medium-low for 2 minutes, then mix by hand to remove large air pockets. The frosting should hold a soft peak without looking greasy or runny.

Use a serrated knife to trim domed tops only if necessary. Set one layer on a cake board and spread about 180 g buttercream evenly over it. Pipe a raised buttercream ring around the edge, then spread 40 g cooled caramel inside the ring. Repeat with the second layer, keeping the caramel away from the exposed edge. Add the final layer upside down for a flat top, checking from several angles that the stack remains centered and vertical.

Spread a thin coat of buttercream over the top and sides with an offset spatula, trapping loose crumbs rather than chasing a perfect finish. Fill obvious gaps between the layers, then smooth with a bench scraper while rotating the turntable. Refrigerate the cake uncovered for 30 minutes, until the coating feels firm and no longer transfers to a fingertip. Keep the remaining buttercream at cool room temperature and stir it smooth before the final coat.

Apply the remaining buttercream in an even final coat and smooth the top and sides. Chill for 20 minutes. Warm the remaining caramel gently until it flows from a spoon but is not hot, then test one drip on the back of the cold cake before finishing the edge and top. Arrange the cooled caramel popcorn in a generous crown immediately before serving. Use a warm, dry knife for clean slices, wiping the blade between cuts and checking that no hard kernels remain.