MAKE THE CAKE: Preheat the oven to 350 degrees. Butter two 9-inch round cake pans. Line the bottoms with parchment paper and butter the paper. Dust the pans with flour, tapping out the excess.

In a medium bowl, whisk the 1 1/2 cups of all-purpose flour with the cake flour, baking soda, baking powder, cinnamon, and salt. In a standing mixer fitted with a paddle, beat the butter with the sugar at medium speed until fluffy, about 3 minutes. Beat in the eggs and vanilla until incorporated. In 3 alternating additions, add the dry ingredients and sour cream, scraping down the side of the bowl between additions.

Scrape the batter evenly into the prepared pans. Bake the cakes on the lower and middle racks of the oven for about 30 minutes, until golden and springy and the edges begin to pull away from the sides of the pan. Transfer the cakes to racks and let cool for 15 minutes, then turn the cakes out and let them cool completely. Remove the parchment paper.

MEANWHILE, MAKE THE FROSTING: In a standing mixer fitted with a paddle, beat the butter until creamy. At low speed, beat in the confectioners' sugar. Increase the speed to high and beat until fluffy, scraping down the side of the bowl, about 2 minutes longer. At low speed, beat in the cream, vanilla, cinnamon, ancho chile powder, and cayenne, then gradually beat in the bittersweet and white chocolate. Scrape down the side of the bowl and the paddle, increase the speed to medium-high and beat until light, 3 minutes longer. Using a wooden spoon, vigorously beat the buttercream for 30 seconds to deflate any air bubbles.

Place one cake layer on a plate and spread with 1 cup of the frosting. Top with the second cake layer and spread the remaining frosting over the top and sides. Refrigerate the cake for at least 30 minutes. Bring to room temperature before serving.

The cake can be refrigerated for up to 3 days. The buttercream can be refrigerated for up to 5 days. Return to room temperature before using.