Make the Cake: Preheat oven to 350 degrees F. Grease and flour a 10-cup or 12-cup Bundt pan.
In a medium bowl, whisk together the flour, baking powder and salt.
Using an electric mixer, beat the butter and sugar on medium speed until light and fluffy, about 3 minutes. Scrape down the sides of the bowl, then add the eggs and the yolks one at a time, beating well after each addition. Scrape down the bowl again, add the vanilla extract, and beat until just incorporated.
Add the flour mixture in three parts, alternating with the milk, beginning and ending with the flour mixture, mixing after each addition until just combined; do not overmix.
Transfer 1 cup of the batter to a small bowl and stir in red food coloring until desired color is reached; set aside.
Transfer another 1 cup of the batter to a separate small bowl and stir in blue food coloring until desired color is reached; set aside.
Spoon the red batter into the prepared pan, then top with the plain white batter, smoothing it into an even layer. Finally, spoon the blue batter into the center of the white layer, leaving some white batter on either sides so it creates a "tunnel".
Bake until a sharp knife or toothpick inserted in the center of the cake comes out with just a few moist crumbs, 50 to 55 minutes. Transfer the pan to a wire rack to cool completely. Gently loosen the sides of the cake from the pan and turn it out onto the rack.
Make the Glaze: In a large bowl, whisk together 4 tablespoons of the milk and the vanilla extract. Add the powdered sugar and whisk until incorporated and smooth. Slowly stir in the white chocolate. The glaze should be thick yet pourable; if the glaze appears too thick, thin it out with additional milk, a tablespoon at a time, until you reach the desired consistency.
Pour the glaze in large, thick ribbons over the crown of the cake, allowing the glaze to spread and drip down the sides of the cake. Allow the glaze to set, about 5 minutes. In small bowls, mix two small batches of powdered sugar with milk until a thick, yet pourable consistency is reached, then dye one red and the other blue. Using a spoon drizzle the red and blue icings over the white. Allow to set for about 10 minutes before serving.
The cake can be kept in an airtight container at room temperature for up to 3 days.