Add the chocolate cake mix, 2 eggs, 1/3 cup oil, and 1 teaspoon vanilla to a large bowl. Stir until no dry cake mix remains and a thick dough forms. Set aside.

In a separate bowl, combine the strawberry cake mix, remaining 2 eggs, remaining 1/3 cup oil, and remaining 1 teaspoon vanilla. Stir until a thick, smooth dough forms.

Scoop about 1 tablespoon of chocolate dough and 1 tablespoon of strawberry dough. Press the two portions together, then gently roll them into a ball while keeping the colors distinct.

Arrange the dough balls about 2 inches apart on the prepared baking sheets. Lightly flatten each one with your fingertips or the bottom of a measuring cup.

Bake for 10 to 12 minutes, or until the edges are set and the centers still look slightly soft.

Let the cookies cool completely before frosting.

To make the buttercream, beat the softened butter in a large bowl for 1 to 2 minutes, or until smooth and creamy.

Gradually mix in the powdered sugar on low speed. Add the vanilla and 1 tablespoon of milk or cream.

Beat the buttercream for 2 to 3 minutes, or until light and fluffy. Add additional milk or cream, a little at a time, if needed to reach a pipeable consistency.

Transfer the buttercream to a piping bag fitted with a plain round tip or no tip at all. Pipe a swirl around each cooled cookie.

Serve immediately, or allow the frosting to sit until it sets slightly on top. To help the frosting set faster, you can also chill frosted cookies in the fridge.