Preheat oven to 350° F. Grease and flour a 10-inch bundt pan.
Whisk together the flour, baking powder, salt and ground pistachios. Set aside.
In the bowl of a stand mixer (or using a hand mixer), cream together the butter and sugar until light and creamy, about 2 minutes. Add eggs one at a time, beating well after each addition. Scrape down the sides and bottom of the bowl as necessary. Mix in pistachio extract and vanilla extract.
Alternate adding the flour mixture and the milk, finishing and ending with the flour mixture. Do not overmix.
Pour batter into prepared pan. Bake at 350° F for 60 to 70 minutes, or until a toothpick inserted into the center of the cake comes out clean. Cool in the pan for 10 minutes, then invert onto a wire rack to cool completely.
Make the icing. Whisk together confectioners' sugar, milk and vanilla extract. Pour over cooled cake and garnish with shelled pistachios. Slice and serve.
Notes
Grind pistachios in a food processor or finely chop them with a sharp knife.Pistachio extract: This may be difficult to find. If you don't want to order it online, I suggest substituting with 1 teaspoon almond extract. It has a flavor that will complement the ground pistachios.Store covered at room temperature for up to 3 days or in the refrigerator for 5 days.Freeze for up to 3 months. Wrap the slices (or whole cake) in a double layer of plastic wrap and store in an airtight container or freezer bag. Defrost in the refrigerator before serving.