Preheat oven to 350 F, placing the rack in the bottom third of the oven.
Spray the bundt pan generously with non-stick spray.
Put the bittersweet and white chocolates in two separate bowls and microwave each small bowl in twenty second increments until just melted. Stir to fully melt. Set aside.
In the bowl of a stand mixer using the paddle attachment, or in a large bowl using a hand mixer, cream together the soft butter and sugar until the butter mixture is light and fluffy. Beat in the vanilla extract and then the eggs, one at a time, scraping down the sides of the bowl between eggs, until the mixture is fully combined.
Sift the flour, baking powder, and salt into the wet ingredients. Beat on low speed to combine.
Scrape two thirds of the batter into a medium bowl and stir in the white chocolate. Add the melted dark chocolate, cocoa powder, and milk into the remaining third of the batter and beat the dark batter until fully combined.
Scrape half of the white chocolate mixture into the prepared pan. Pour cocoa batter over the top of the light dough. Then add the remaining batter on top of the chocolate cake batter. Using swooping motions, drag the blade of a knife through the batter to marble the vanilla and chocolate part.
Place the pan in he heated oven and bake 45-60 minutes, or until a cake tester inserted into the center of the cake comes out clean.
Place on a wire rack let the cake cool 10 minutes, before unmolding and letting it fully cool removed from the pan. Dust with powdered sugar to serve.
This cake is best served with a cup of coffee. It keeps well, wrapped in plastic wrap, for up to three days.