Cake: Preheat oven to 350 degrees. Butter a 9‐by‐13‐inch baking pan. Beat butter with sugar, baking powder, and salt on medium speed until light and fluffy, about 3 minutes. Add egg and yolks, one at a time, beating to combine after each addition and scraping down sides as needed. Beat in both vanillas, then flour in three additions, alternating with milk and beginning and ending with flour. Scrape batter into prepared pan, smoothing top with an offset spatula.

Bake until a tester inserted in center comes out with a few moist crumbs, 35 to 40 minutes. Transfer pan to a wire rack; let cool completely.

Buttercream: Place chocolate in a heatproof mixing bowl. In a small saucepan, bring cream and corn syrup to a boil. Pour mixture over chocolate; let stand 2 minutes, then stir until smooth. Let cool 30 minutes.

Beat, gradually adding butter, until all has been added and buttercream is silky and smooth. Continue beating, scraping down bowl as needed, until light and fluffy, about 1 minute more. Makes about 3 cups. Use immediately.

Assembly: Spread buttercream evenly over cake. (Leftovers can be refrigerated, wrapped in plastic, up to 3 days.)