Position racks evenly in the oven and preheat to 350 degrees F. Line 3 baking sheets with parchment paper.
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.
Whisk the flour, baking powder, and salt together in a large bowl.
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.
While mixing slowly, add the dry ingredients to the wet, in 2 additions, mixing just until absorbed. Fold in the peanuts and chocolate pieces.
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.
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.
Store cookies in a tightly sealed container for up to 3 days.