Preheat the oven to 350 degrees F and line 2 baking sheets with parchment paper; set aside. Place the pistachios on a rimmed baking sheet and toast in the oven, shaking the baking sheet occasionally, until fragrant, about 10 minutes. Remove to a cutting board and allow to cool slightly. Roughly chop, then cool completely.
In a mixer, beat together the butter, vanilla, salt, eggs and egg yolk. Add the flour, sugar and baking powder and mix on low until a rough dough begins to form. Add the pistachios and chocolate chips and continue to mix. If the dough seems dry, add some cold water a teaspoon at a time. If the dough is too wet, add a pinch of flour.
Turn the dough out onto a lightly floured cutting board and knead gently to make sure the chocolate chips and pistachios are evenly distributed. Divide the dough in half, and shape each into a 10-by-3-inch log. Place onto one of the prepared baking sheets and bake until golden, about 30 minutes. Reduce the oven temperature to 300 degrees F. Let the biscotti rest until cool enough to handle, about 10 minutes.
Slice the logs into 3/4-inch pieces, and arrange them on the baking sheets, cut-side down. Bake until toasted and golden on top, about 25 minutes. Flip them over and continue to bake until dry and just barely golden brown, another 25 minutes. Transfer to a rack to cool completely, about 15 minutes.
Biscotti will keep in an airtight container for 2 weeks.