In the bowl of a stand mixer fitted with the paddle attachment, beat the butter on medium speed until pale and creamy, 5 to 8 minutes. Add the vanilla and beat for an additional 30 seconds.
Reduce the speed to low and slowly add the sugar in three additions, scraping down the sides in between, as necessary.
Increase the speed to medium and beat until fluffy. Add the milk or cream until the desired consistency is reached, if needed.
Remove half of the buttercream into a clean bowl. To the buttercream in the stand mixer, add the food color little by little until the desired color is reached.
To each of the buttercream colors, divide the Oreos and stir with a small spoon until combined.
Notes
Assembly
Level the cakes with a serrated knife or cake leveler (like this). Wrap each cake with plastic wrap and freeze for at least 4 hours (overnight preferred).
Place a cake stand on a rotating cake stand. Place a dollop of buttercream onto the cake stand. Center one frozen cake layer onto the cake stand. Add a generous layer of pink buttercream and spread, using an offset spatula, to the edge to form a smooth and even layer.
Place the second frozen cake layer top-side down onto the buttercream layer. Fill two piping bags with each of the buttercream.
With one hand controlling the rotating cake stand, pipe three rows of white buttercream from the bottom up. Repeat with three rows of pink buttercream and finally, three rows of white buttercream. Pipe the remaining white buttercream onto the top of the cake and spread until covered using an offset spatula.
Holding a bench scraper steady against the frosting with one hand and rotating the cake stand with the other hand, scrape off the excess frosting until smooth.