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Preheat the oven to 375°. Line several large cookie sheets with parchment paper. In a microwave oven or in a medium heatproof bowl set over a saucepan of simmering water, melt the unsweetened and bittersweet chocolates. Let cool slightly.
In a medium bowl, whisk the flour with the baking soda and salt. In a large bowl, using an electric mixer, beat the butter at high speed until light and fluffy. Add the brown and granulated sugars and the vanilla and beat at medium speed until combined. Beat in the peanut butter, then the melted chocolate and the egg. At low speed, gradually beat the dry ingredients until incorporated into the cookie dough.
Using your hands, roll the dough into 48 one-inch balls and arrange 2 inches apart on the prepared baking sheets. Using the tines of a fork, flatten the balls to a 1/2-inch thickness. Bake 1 sheet of cookies at a time for about 12 minutes, or until the cookies are semifirm to the touch when lightly pressed. Rotate the cookie sheet halfway through baking for even browning. Slide the parchment paper with the cookies onto a wire rack to cool.
The cookies can be stored between sheets of wax paper overnight in an airtight container.