Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper.
Using a stand mixer, beat the butter and the sugar together until fluffy, about 5 minutes. Add the eggs, one at a time, incorporating each egg fully before adding the next. Add the vanilla. In a small bowl, whisk together the flour, baking powder and salt. With the mixer on low, stir the flour mixture into the butter mixture until just combined. Stir in the dried cherries and pistachios. Cut the dough in half and place the dough on the prepared baking sheet. Shape into 2 logs, each about 12 inches long.
Bake until just starting to brown at the edges, about 20 minutes. Cut each log on the diagonal into 1/2-inch-wide slices. Place the slices back on the baking sheet. Bake on the first side for 5 minutes. Remove, turn the biscotti over to the second side and bake until the biscotti are crisp, an additional 5 minutes. Let cool.