Preheat the oven to 350 degrees.
Lay the pistachios on a cookie sheet in a single layer. Bake them for 10 minutes or until they're lightly toasted. Remove them from the oven.
In a mixer bowl, with a mixer, beat the butter until it's light and fluffy. With the mixer still on, gradually add the eggs, sugar, and vanilla; mix until they're creamed. (light and fluffy)
In a separate bowl, mix the flour, baking powder, and salt until well blended.
Add these dry ingredients to the wet ingredients that are still in the mixer bowl to make a dough.
Switch the mixer's beater(s) to the dough beater(s). Mix the dough until it's smooth.
Using a wooden spoon, mix in the pistachios until they're evenly distributed.
Put the dough on a lightly floured surface and cut it in half. Roll each half into a log, that's each 12 inches long by 1-inch high.
Place the logs onto an ungreased cookie sheet; bake them for 35 minutes or until the bottoms of the logs are light brown. Let the logs cool for 5 minutes; then place them on a cutting board.
Slice each log on a diagonal into 12 1-inch thick pieces.
Place the cookies back on the cookie sheet in a single layer; bake them for 5 minutes.
Turn each one over and bake them for another 5 minutes.
Store the cookies in an airtight container.