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Position racks evenly in the oven and preheat to 350 degrees F. Line 3 baking sheets with parchment paper.
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Melt the butter over medium heat, swirling the pan occasionally. Continue to cook until the butter browns and gets a nutty aroma, about 5 minutes. Let cool slightly.
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Whisk the flour, baking powder, and salt together in a large bowl.
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Beat the eggs in a medium bowl with an electric mixer until light and pale yellow, about 2 minutes. Gradually add the sugar while beating. Then slowly add the butter and vanilla until evenly mixed, about 30 seconds more. Add the peanut butter and mix until combined.
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While mixing slowly, add the dry ingredients to the wet, in 2 additions, mixing just until absorbed. Fold in the peanuts and chocolate pieces.
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Divide the dough evenly into thirds, and put each portion in the center of a baking sheet. Shape the dough with slightly wet hands into logs about 2-inches wide and 15 inches long. Bake until set and brown around the edges, about 25 to 30 minutes. (For even baking take care to rotate the pans-- top to bottom and front to back--about half way though.) Cool logs on the baking sheets for about 10 minutes.
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Lower the oven temperature to 325 degrees F. Carefully transfer the logs to a cutting board. Cut logs crosswise, with a long serrated knife at about a 45 degree angle, into 1/2-inch thick cookies. Place cookies cut side down on the baking sheets. Bake until crisp, about 8 minutes. Flip the cookies over and bake until golden brown, about 8 minutes more. Cool biscotti on the baking sheets. Serve.
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Store cookies in a tightly sealed container for up to 3 days.