In a large bowl, cream together butter and sugar. Beat in flour until dough comes together. It will look shaggy at first, but will gather together.
Dump dough onto a clean work surface and knead until fully cohesive and smooth. Divide dough in half and place into separate bowls.
In one bowl, knead in the vanilla extract. In the other, knead in the cocoa powder.
Divide the halves into half again so you have 4 pieces. Roll each piece into a log about 12 inches long. Use two straight edges to press the dough so that they will form squares when cut.
Lay one vanilla log next to a chocolate log and gently squeeze together to stick. Square up the edges again. Place the remaining vanilla log on top of the chocolate log and vice versa, creating a square. Square up the edges once more. Wrap tightly in plastic wrap, preserving the shape, and chill in the refrigerator for 30 minutes.
Preheat the oven to 350°F and line two baking sheets with parchment or reusable liner.
Remove dough from the refrigerator, unwrap, and slice into 1/4-inch pieces. Arrange on the baking sheets. They will not spread much, so they can be placed close together.
Bake for 8-10 minutes, or until vanilla cookie parts are very lightly golden, but not browned on the edges. Let cool on the baking sheets 2 minutes, then transfer to a wire rack to cool completely.