Preheat oven to 350°F (175°C) and grease a 9-inch cake pan.
Beat butter and sugar until light and fluffy.
Add eggs one at a time and mix in vanilla.
Whisk flour, baking powder, baking soda, and salt.
Gradually mix dry ingredients with buttermilk into the butter mixture.
Fold in crushed pistachios.
Pour batter into pan and bake 30–35 minutes; cool completely.
Whip heavy cream with powdered sugar, fold in pistachio paste and vanilla.
Spread frosting over cooled cake and chill 30 minutes.
Garnish with chopped pistachios, slice, and serve.