Vanilla Dough - In a stand mixer fitted with the paddle attachment, add the softened butter, granulated sugar, powdered sugar, and salt. Gradually increase to medium-high speed (setting 6) and beat for 1 minute. Scrape down the bowl's sides and paddle with a spatula. Beat until fluffy and pale in color, 30 seconds.Scrape down the bowl. Add egg yolks and vanilla extract, and beat on medium-high speed (setting 6) until combined, about 30 seconds.Scrape down the bowl. Turn the mixer to the lowest speed setting (stir) and gradually add flour. Mix until combined and clumpy, about 45 seconds. Scrape down the bowl halfway through. Transfer dough to a bowl, cover, and set aside. Save the mixing bowl to use for the chocolate dough.
Chocolate Dough - Add the softened butter, granulated sugar, powdered sugar, and salt to the mixer bowl. Gradually increase to medium-high speed (setting 6) and beat for 1 minute. Scrape down the bowl's sides and paddle with a spatula. Beat until fluffy and pale in color, 30 seconds.Scrape down the bowl. Add egg yolks and vanilla extract, and beat on medium-high speed (setting 6) until combined, about 30 seconds.In a small bowl, whisk together the flour and cocoa powder. Scrape down the bowl. Turn the mixer to the lowest speed setting (stir) and gradually add the flour and cocoa mixture. Mix until combined and clumpy, about 45 seconds. Scrape down the bowl halfway through.
Roll Out the Dough - Transfer the chocolate dough to a piece of parchment paper. Shape into a 1" thick rectangle. Dust the rolling pin and dough with flour, using more as needed to prevent sticking. Roll into an 8" by 6" rectangle. Set aside. Use your hands to make the edges straight. Repeat the process with the vanilla dough using a new piece of parchment paper.Stack the dough on top of each other. Roll into a 12" by 8" rectangle, dusting with flour as needed and occasionally flipping it over. Use your hands to straighten the edges. Starting on the long side of the dough, roll it into a tight cigar-like shape. Use the parchment paper to help roll the dough over itself.
Chill the Dough - Cut the dough logs down the middle into two pieces. Wrap each one tightly in plastic wrap. Refrigerate until firm, about 1 hour, or freeze for 30 minutes.
Preheat the Oven - Set the oven rack to the lower-middle and upper-middle positions. Heat oven to 325ºF (163ºC). Line two baking sheets with parchment paper. If you are only baking one tray or want crisper cookies, bake them in the middle of the oven.
Slice the Cookies - Allow the cookie dough to sit on the counter while the oven preheats, about 15 to 20 minutes before slicing. Slice each log into ¼" thick pieces, and place them 1" apart on the baking sheets.
Bake - Place the pans in the oven and bake for 7 minutes, then switch the pan positions and rotate. Continue to bake until the edges set, and the vanilla turns light golden, about 8 to 10 minutes.Leave the cookies on the baking sheet for 10 minutes, then transfer them to a wire rack to cool completely.
Notes
Make ahead: The cookie dough can be wrapped and stored in a resealable plastic bag in the refrigerator for up to 1 week or the freezer for up to 2 months. Defrost, and bring to a sliceable temperature before using.
Storing: The cookies can be stored in an airtight container for up to 1 week at room temperature or freezer for up to one month.