Preparation
Position rack in the center of the oven; preheat oven to 350°F. Line a baking sheet with parchment paper.
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.
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.
Drop by rounded tablespoonsful 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.