Steps:
- Whisk 1/13 cups of flour and 3/4 tsp. salt in a medium bowl.
- Whisk cocoa powder, remaining 1 cup of flour, and remaining 3/4 tsp. salt in another medium bowl.
- These are the bases for your chocolate and vanilla doughs.
- Beat butter, granulated sugar, and powdered sugar in the bowl of a stand mixer on medium-high speed until light and fluffy, about 4 minutes.
- Add egg and vanilla and beat until smooth.
- Divide mixture between the 2 bowls of dry ingredients (about 1 cup each).
- Scrape vanilla mixture back into stand mixer bowl (save the mixing bowl) and beat on low speed just until combined. Return to reserved bowl.
- Repeat process with chocolate mixture.
- Arrange 2 large sheets of parchment paper on a work surface. Dollop 1/4 of chocolate dough in the center of each sheet and pat into rough 6x2" rectangles.
- Dollop 1/4 of vanilla dough on top of each chocolate slab and pat into rectangles the same size and shape so that you have 2 layers each.
- Repeat entire process so you have 4 alternating layers .
- Tightly press stacked dough into cylinders about 1 1/2" wide and 8" long, using the parchment to help you.
- Wrap logs in plastic wrap and hill until very firm, at least 2 hours.
- Place racks in upper and lower thirds of oven, preheat to 350 degrees.
- Working one at a time, unwrap dough and brush with egg. Carefully sprinkle surface with sanding sugar and roll logs in sugar to coat well (really press dough into sugar so it sticks).
- Slice into rounds a generous 1/4" thick, rotating after every few cuts to keep slices round.
- Arrange cookies on a parchment-lined baking sheets, spacing 2" apart. Bake, rotating baking sheets top to bottom and front to back halfway through, until edges are just set, 12-14 minutes.
- Let cool on baking sheets.
- DO AHEAD: Dough can be made 3 days ahead; keep chilled. Cookies can be baked 5 days ahead; store airtight at room temperature.
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