In a stand mixer fitted with the paddle attachment (or a handheld mixer with an eggbeater or whisk attachment), beat butter on high speed until as pale as ivory and very fluffy, at least 5 minutes. Scrape sides and bottom of bowl twice while beating.
Turn the mixer to low and gradually add sugar. When combined, scrape bowl again, then slowly drizzle in milk and vanilla; mix until combined. Mix in pistachio paste. If you'd like the frosting to be greener, in a small bowl, blend yellow and blue food colorings to make a green tint, scrape into frosting, and mix to blend. Scrape bowl again.
Turn the mixer to high and beat the buttercream until extremely fluffy, 8 to 10 minutes. "It should look almost like whipped cream, and should lose the waxy sheen that butter has."
Make ahead: Up to 3 hours at room temperature. Although you can make buttercream a few days ahead and chill, it will take a while to return it to room temperature (about 3 hours) and re-whip to the proper cloudlike texture--much longer than just making it fresh.