Position rack in the center of the oven; preheat to 350 degrees F. Line a large baking sheet with parchment paper or a silicone baking mat.
Whisk all-purpose flour, whole-wheat flour, cocoa powder, baking soda and salt in a medium bowl. Beat egg whites in a large bowl with an electric mixer until foamy, about 1 minute. Beat in granulated sugar in a slow, steady stream. Scrape down the sides, then beat in brown sugar 1 tablespoon at a time. Beat until smooth, about 3 minutes. Beat in vanilla and melted chocolate. Stir in the dry ingredients with a wooden spoon until just incorporated.
Drop the batter by tablespoonfuls onto the prepared baking sheet, 1 1/2 inches apart.
Bake the cookies until flat yet springy, with slightly cracked tops, 10 to 12 minutes. Cool on the pan for 5 minutes, then transfer to a wire rack to cool completely. Let the pan cool for a few minutes before baking another batch; replace parchment paper if torn or scorched.
