In a medium bowl, sift cake flour, cocoa, salt, baking powder, and baking soda together.

In the bowl of an electric mixer fitted with a paddle attachment, or in a large bowl with a hand-held mixer, mix the oil and the sugar until well-blended. Add eggs and egg yolk one at a time, mixing after each. With the mixer on low, slowly add all of the red food coloring, being very careful not to splash. Add peppermint extract.

Add flour mixture in three additions, alternately with the buttermilk and ending with the flour. Mix until just combined, scraping bowl down as needed; don't overmix.

To prepare the cheesecake filling, beat the cream cheese, sugar, egg, flour and peppermint extract until smooth.

Pour half of the red velvet batter into the prepared bundt pan. Spoon cheesecake filling over the batter, leaving a 1 inch border. Use a knife to gently swirl some of the batter into the filling. Top with remaining red velvet batter.

Bake until a toothpick inserted into the center comes out with a few moist crumbs or the top springs back when pressed gently, about 1 hour. Let cool in the pan 15 minutes, then transfer to a wire rack to cool completely.

Combine the melted white chocolate and heavy cream, then stir in the powdered sugar and peppermint extract. If it's too thin, let it sit for 5-10 minutes. If it's too thick, add more cream a teaspoon at a time.

Gently pour glaze over the cooled cake. Top with crushed candy canes and let it set for at least 15 minutes before serving. Store cake in the fridge for up to 5 days.