Preheat the oven to 375 degrees F. Line a large cookie sheet or 2 cookie sheets with parchment paper or a baking mat.
In a large mixing bowl, beat the butter for 30 seconds at high speed to soften it and break it up. Add the sugar, baking powder, and salt, and beat the mixture until combined, scraping down the sides of the bowl as needed.
Add the eggs and vanilla to the bowl and beat until well combined, an additional 1-2 minutes.
Add the lemon peel and the flour and stir until combined. (You can continue to use a stand mixture here but I find it easier to just use a rubber spatula.) Gently stir in the pistachio nuts.
On a lightly floured surface, divide the dough into 3 equal portions. Shape each portion into an 8-inch long loaf and flatten to about 2 ½ inches wide. Place the loaves at least 3 inches apart from each other on the prepared cookie sheet/s.
Bake the cookie loaves for 20-25 minutes, or until the loaves are golden brown and cracked on the top. (Note, if you spread the loaves across 2 cookie sheets, you may want to rotate the sheets halfway through baking to ensure even baking.) Cool on the cookie sheet for 30 minutes.
Once cooled, transfer the loaves to a large cutting board and cut each loaf diagonally into ½ inch thick slices. Lower the oven to 325 degrees F. Return the slices to the prepared cookie sheets and place them cut sides down. Bake for 8 minutes, then turn and bake for 8-10 minutes more until crisp. Transfer the lemon biscotti cookies to a wire rack to cool.
Meanwhile, make the lemon glaze. In a small mixing bowl, stir together the powdered sugar and shredded lemon peel. Add in enough milk or lemon juice to thin out the icing to your desired consistency.
Once the lemon biscotti cookies have fully cooled, dip the ends into the lemon icing, add a lemon glaze drizzle to each cookie. Allow the cookies to sit at room temperature until the icing hardens, about 20 minutes.