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Whisk whole-wheat pastry flour, all-purpose flour, cocoa, baking soda and salt in a medium bowl.
Blend brown sugar, granulated sugar and butter in a large bowl with an electric mixer on medium speed until combined and still a little grainy, about 5 minutes. Beat in egg white and vanilla until smooth. Turn the mixer off; add the flour mixture then beat at low speed until well combined (the dough will still be crumbly). Gather the dough into a ball using your hands.
Lightly dust a work surface with flour, turn the ball of dough out onto it and form into an 8-inch-long log, about 2 inches in diameter. Wrap in plastic wrap and refrigerate until firm, at least 2 hours or up to 2 weeks.
When ready to bake, position racks in the upper and lower thirds of the oven; preheat to 350°F. Line 2 large baking sheets with parchment paper or silicone baking mats.
Unwrap the log and slice into 1/4-inch-thick rounds, rolling the log as you slice to prevent it from flattening. Place the cookies 1 inch apart on the baking sheet. Bake, switching the pans back to front and top to bottom halfway through, until just beginning to crack along the edges, 10 to 12 minutes. Cool on the baking sheet for 2 minutes, then transfer to a wire rack to cool completely, about 20 minutes.