If making your own pistachio paste, make it ahead of time so it has a chance to cool. You can also make it up to 4 weeks in advance and store in the fridge. Alternatively, you can find pistachio paste online if you'd rather not make it at home.
Preheat oven to 350°F / 180°C (160°C fan-assisted) and grease two 8-inch cake pans. I also like to add parchment paper circles on the bottom to avoid sticking.
Cream the butter, sugar, and pistachio paste together until light and fluffy.
Beat in the eggs one at a time until incorporated.
Sift the dry ingredients together, then add half of the mixture to the cake batter. and mix in slowly.
Add the wet ingredients, then mix until completely incorporated.
Stir in the other half of the dry ingredients until there are no more lumps - but be careful not to overmix.
Fold the chopped pistachios into the batter.
Divide the cake batter between the prepared cake pans and bake for 30-35 minutes, or until a cake tester inserted in the middle comes out with a few moist crumbs.
Allow to cool in the pans for 15 minutes before removing onto a wire rack to cool completely.
Once cooled, slice each cake layer in half horizontally to make four thinner layers. I like to use a cake leveler for this, but a serrated knife works well too.
Make the Italian meringue buttercream while your cake is cooling (or bring it to room temperature if you prepared it in advance). Add the pistachio paste and whip until combined and smooth.
Fill and frost the pistachio cake, and enjoy!
Notes
If you are buying pistachio paste rather than making your own, make sure you get pistachio paste or pistachio butter - NOT pistachio cream. The ingredients should be 100% pistachios. See the post above for more info on this. Want a different frosting? You can turn most recipes into a pistachio frosting by adding a few tablespoons of pistachio paste right at the end. I have made this cake many times with a quick Russian buttercream frosting, which has the silky texture but only takes 10 minutes to make.