1. Place eggs in small saucepan, cover with 1 inch of water, and bring to boil over high heat. Remove pan from heat, cover, and let sit for 10 minutes. Meanwhile, fill small bowl with ice water. Using slotted spoon, transfer eggs to ice water bath and let stand 5 minutes. Separate yolks from whites; discard whites. Press 1 yolk through fine-mesh strainer into small bowl; set aside. Repeat with remaining yolk, pressing into separate small bowl (keep yolks separate).

2. For Vanilla Dough: In bowl of stand mixer fitted with paddle attachment, beat butter, granulated sugar, salt, and 1 cooked egg yolk on medium speed until light and fluffy, about 4 minutes, scraping down sides of bowl and beater with rubber spatula as needed. Turn mixer to low, add vanilla, and mix until incorporated. Stop mixer; add flour and mix on low speed until just combined, about 30 seconds. Using rubber spatula, press dough into cohesive mass. Transfer dough to large plate and set aside while preparing chocolate dough.

3. For Chocolate Dough: In now-empty bowl of stand mixer fitted with paddle attachment, beat butter, granulated sugar, salt, and remaining 1 cooked egg yolk on medium speed until light and fluffy, about 4 minutes, scraping down sides of bowl and beater with rubber spatula as needed. Turn mixer to low, add vanilla, and mix until incorporated. Stop mixer; add flour and cocoa; mix on low speed until just combined, about 30 seconds. Using rubber spatula, press dough into cohesive mass.

4. To form Checkerboards: Divide vanilla and chocolate doughs in half. Roll each portion (4 pieces total) into 3 x 5-inch rectangle. Place sheets of vanilla dough on top of sheets chocolate dough and gently press to seal. (You should have 2 sandwiched masses of dough.) Chill dough for 30 minutes.

5. Slice each sandwiched dough lengthwise into 4 equal strips. Turn one strip onto its side, so vanilla and chocolate is side by side. Turn second strip over and place on top of first strip, creating a checkerboard pattern; press gently to adhere. Wrap gently in plastic wrap and chill for 1 hour. Repeat with remaining strips (you should have 4 logs total).

6. Adjust oven racks to upper-middle and lower-middle positions and heat oven to 350°F. Line two rimmed baking sheets with parchment paper. Using sharp chefs knife, cut 2 dough logs into 1/4-inch-thick slices, rotating dough so it wont become misshapen from weight of knife. Place cookies 1 inch apart on baking sheets.

7. Bake until center of cookies are pale golden brown, with edges slightly darker, about 15 minutes, rotating baking sheets front to back and top to bottom halfway through baking. Cool cookies on baking sheet 5 minutes; using wide metal spatula, transfer cookies to wire rack and cool to room temperature. Repeat steps 6 and 7 with remaining 2 logs of dough. Store cooled cookies between sheets of parchment or wax paper in airtight container for up to 1 week.

Makes about 80 cookies