Move rack to center of the oven; preheat to 350°F. Line a large baking sheet with parchment paper.
In a medium bowl, mix all-purpose flour, whole-wheat flour, cocoa powder, baking soda and salt.
In a separate large bowl, beat egg whites with an electric mixer about 1 minute, until foamy. Beat in granulated sugar in a slow, and steady stream. Beat in brown sugar one tablespoon at a time. Beat about 3 minutes; until smooth. Beat in vanilla and melted chocolate. Stir in the dry ingredients with a wooden spoon until just incorporated.
Drop batter by tablespoonfuls onto the baking sheet about 1 1/2 to 2 inches apart.
Bake the cookies until flat yet springy, with slightly cracked tops, about 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.