Beat 1 cup butter and 3/4 cup sugar until light and fluffy, about 3 minutes. Beat in 1 egg and 1 teaspoon vanilla. Add 2 cups flour and 1/4 teaspoon salt; mix just until dough forms. Shape into a disk, wrap in plastic, and refrigerate while making chocolate dough.

Beat remaining 1 cup butter and 3/4 cup sugar until light and fluffy. Beat in remaining egg, vanilla, and cocoa powder. Add remaining 2 cups flour and 1/4 teaspoon salt; mix until crumbly. Add milk and mix until dough comes together. Shape into a disk and wrap in plastic.

Refrigerate both doughs until firm, at least 1 hour or up to 2 days. Dough must be cold but pliable for rolling.

On parchment paper, roll vanilla dough into a 10x12-inch rectangle about 1/4 inch thick. Repeat with chocolate dough on separate parchment. If dough cracks, let it warm slightly at room temperature.

Flip chocolate dough onto vanilla dough, peeling off parchment. Trim edges to align. Starting from a long side, roll tightly into a log, using parchment to guide. Wrap in plastic and refrigerate until very firm, at least 2 hours or overnight.

Preheat oven to 350 degrees F. Unwrap log and slice into 1/4-inch rounds. Place 2 inches apart on parchment-lined baking sheets. Bake 10-12 minutes until edges are set and vanilla dough looks dry, not wet. Cool on sheets 5 minutes, then transfer to wire racks.