To Prepare the Pistachio Paste:
Place the shelled pistachios into a food processor or blender and pulse until they are very finely ground.
Add the heavy cream and pulse again until a thick paste forms. Set the pistachio paste aside.
To Make the Buttercream:
Soften the butter at room temperature until it is lightly softened. Place the butter into the bowl of a stand mixer fitted with the flat beater attachment.
Beat the butter on medium to high speed for 7 to 9 minutes, until the butter is fluffy and white in color. Be sure to scrape down the sides of the bowl frequently to ensure even mixing.
Add the pistachio paste and vanilla extract to the butter. Beat for a few minutes, scraping down the sides of the bowl, until the mixture is well combined.
Gradually add the confectioner’s sugar, mixing on low speed until the sugar is incorporated. Once all the sugar is added, increase the speed to high and beat for 3 to 4 minutes until the buttercream is light and fluffy.
If desired, add a few drops of green food coloring to achieve a more vibrant color.
If your buttercream is not fluffy enough, refrigerate it for 30 minutes, then beat again until it reaches the desired consistency.
Storage:
Refrigeration: If not using immediately, keep the buttercream well covered with plastic wrap. It can be stored in the refrigerator for up to 1 week. Before using, allow the buttercream to soften at room temperature.
Freezing: To freeze, place the buttercream in a sealed plastic container. It can be frozen for up to 1 month. Thaw overnight in the refrigerator, then bring to room temperature before use.