Preheat the oven to 350 degrees F. Spray four 8-inch square baking pans with nonstick baking spray.
Add the granulated sugar and vanilla bean paste to a small bowl and whisk to combine.
Add the cake flour, baking powder, baking soda and salt to the bowl of a stand mixer fitted with the paddle attachment. Stir on low to combine. Add the vanilla bean sugar to the bowl and stir until well combined. Add the butter and mix on medium-low speed until you have a fine crumb texture, about 3 minutes.
Whisk together the eggs, sour cream and oil in a small mixing bowl until smooth. Add the egg mixture to the flour mixture and beat on medium speed until just combined. Be sure to scrape down the sides and bottom of the bowl as needed to make sure everything is incorporated. Slowly add the milk and mix on low until just combined. (The batter will be very liquid.) Remove a quarter of the batter (15 ounces or scant 2 cups) to one of the prepared baking pans and set aside. Remove another quarter of the batter to a mixing bowl. Add a couple drops of red gel food coloring to the mixing bowl and stir to combine until you have a deep red color. Scrape this batter into another prepared baking pan. Add about half the amount of red gel food coloring to the remaining batter in the mixing bowl and mix until you have a medium pink color. Divide this pink batter between the two remaining prepared cake pans. (There should be about 15 ounces in each.)
Bake until a toothpick inserted in the center of a cake comes out clean and the cakes have not turned golden brown, 12 to 18 minutes. Let cool completely in the pans on a wire rack.
Tint 1/2 cup of peppermint buttercream frosting with red gel food coloring to get a deep red color, and set aside 1/2 cup of white. Tint the remaining frosting a light pink color. Place the red frosting in a piping bag fitted with a flat tip for wide stripes, and the white frosting in a piping bag fitted with a flat tip that has 4 to 5 holes in it, for parallel thin stripes.
Trim each cake to about 7-inches square so any browned or uneven edges are removed. Cut each cake into at least six 1-inch strips. (If you didn't have to trim your cake much you might get more; this will just be bonus cake snacks for you.)
Place four strips of cake side-by-side, alternating pink and white. Spread about 1/4 cup pink frosting atop these and spread in an even layer. Place one red stripe atop each pink one, and one pink stripe atop each white one. Spread another 1/4 cup pink frosting atop this layer. Repeat topping the red stripes with pink ones, and the pink strips with white ones. Add another 1/4 cup pink frosting and spread evenly. Repeat topping the pink stripes with red ones and the white stripes with pink ones. Spread 1/4 cup pink frosting in an even layer and place a final layer of pink strips atop the red ones and white strips atop the pink ones. Frost the entire outside of the cake with a thin even layer of pink frosting for a crumb coat. Refrigerate until set, about 30 minutes, then cover the outside with a thicker layer of pink frosting. Use a bench scraper to smooth out the sides and make the angles crisp.
Beginning with the red buttercream and working crosswise on the cake, pipe three evenly-spaced straight lines up one side of the cake, across the top, and down the opposite side. Switch to the white buttercream and pipe two additional sets of lines, running in between (and parallel to) the red lines.
Now working lengthwise on the cake, pipe two evenly spaced straight lines with the red buttercream up one side, across the top, and down the other side of the entire cake. Follow with one set of white lines in the center.
Switch back to the red buttercream and pipe 2 stripes running horizontally around the sides of the cake. Switch back to the white and pipe a set of white stripes running between the red.
Combine the butter, confectioners' sugar and salt in the bowl of a stand mixer fitted with a paddle attachment (or in a large bowl if using a handheld electric mixer). Mix on low speed until mostly incorporated. Add the peppermint extract, then increase the speed to medium-high and beat until smooth. Use immediately, or refrigerate in an airtight container up to 3 days. Allow to come to room temperature before using, then beat on medium-high speed until smooth.