Make dough in a stand mixer or with a hand mixer: Combine butter, sugars, and salt in the bowl of your stand mixer and beat until creamy. If you began with cold butter in a stand mixer, this will take a couple minutes and require you to scrape down the bowl a few times. Once mixture is thoroughly combined, add egg yolk (save the egg white for later) and vanilla and beat until combined. Add 1 1/2 cups (195 grams) of the flour and beat until it disappears into a smooth dough.
Both methods: Divide dough in half. [Each half will weigh 270 to 275 grams.] Leave one half in the mixing bowl or food processor. Add 1/4 cup (35 grams) remaining flour to it and pulse/mix until just combined. Scrape out the mixing bowl or food processor and scoop out this vanilla dough, setting it aside. Add the second half of the dough to the mixing bowl or food process and add the cocoa powder. Mix until evenly combined. This is now your chocolate dough.
Heat oven: To 350°F (176°C)
To checkerboard the doughs:
To finish cookies: Line a large baking sheet with parchment paper, or discarded parchment from your cookie slabs. Beat reserved egg white until just loosened. Gather your sanding sugar(s).
Unwrap first checkerboarded log. Brush log with a thin coat of egg white and roll or sprinkle in sanding sugar. [Normal people choose one color per log. I cut each log into quarters and did each quarter in a different color.] Slice sugar-coated log into 1/4-inch-thick cookies, and arrange on prepared baking sheet with 1-inch space between them (they expand slightly). Repeat with second log, creating more cookies.
Bake cookies: For 10 to 12 minutes, or until golden brown underneath. Let cookies rest on tray for 5 minutes, so they firm up a little, before transfer to a cooling rack to finish cooling and crisping up.
Do ahead: Baked, cooled cookies keep for 3 weeks in a tin at room temperature.
To spiral the doughs: Roll your chocolate and vanilla dough slabs each to 1/8-inch thick and stack them, patting them gently together. Chill in the freezer for just 5 to 10 minutes, until cold but not hard. Divide dough into two long, equal widths. With the vanilla dough underneath/forming the outer ring, roll each half of dough into a tight spiral. [I forgot and did the opposite, but always think it looks better with vanilla on the outside.] Wrap each spiral in parchment, pressing the dough into a firm log. Chill in freezer for 15 minutes, until solid but not frozen hard. Continue from “To finish cookies:” instructions above.
To marble the doughs: Roll your chocolate and vanilla dough slabs each to 1/8-inch thick and stack them, patting them gently together. Cut into two stacked sections and squeeze each into a rough log, fold it over, and mash the log, kneading it once or twice. Press kneaded mounds together and form them back into a long log shape. Wrap tightly with parchment, pressing the dough into a firm log. Chill in freezer for 15 minutes, until solid but not frozen hard. Continue from “To finish cookies:” instructions above.