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Position rack in the center of the oven; preheat oven to 350°F Line a baking sheet with parchment paper.
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Place the chocolate in a small bowl and microwave on High 15 seconds, stir, then continue microwaving in 10 second increments until halfway melted. Remove from the microwave and continue stirring until completely melted.
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With an electric mixer on high, beat the egg whites in a large bowl until foamy, about 30 seconds. Beat in both sugars until creamy, about 1 minute. Beat in the melted chocolate. Remove the beaters and stir in the flour, whole-wheat flour, cocoa powder, baking soda, and salt all at once with a wooden spoon, just until moistened.
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Drop by rounded tablespoonfuls onto the prepared baking sheet. Bake until puffed, cracked, and dry, about 15 minutes. Cool cookies on the baking sheet for 2 minutes, then cool completely on a rack. Cool the baking sheet for 5 minutes before baking additional batches. Store in an air-tight container for up to 5 days.