This frosting begins with a pistachio pastry cream. Bring the half & half and pistachios to boil in a small heavy saucepan. Remove from heat, cover, and let stand for 1 hour.
Whisk the sugar, egg yolks, and flour together vigorously in a medium bowl until the color begins to look pale yellow.
Place the saucepan with the half & half mixture back over medium heat and bring to a simmer. Very gradually whisk the hot mixture, nuts and all, into the sugar and yolk mixture.
Return the combined mixture to the same saucepan and place over medium heat. Then whisk constantly until it begins to bubble thickly, about 3 to 5 minutes. (The thickening will happen all of a sudden, so avoid any distractions during this step to avoid burning the pastry cream.)
Remove the saucepan from the heat and mix in the vanilla extract. Let the mixture cool briefly before transferring it to the food processor and blending until the nuts are very finely chopped and distributed evenly throughout the pastry cream.
Transfer to a bowl and cover with plastic, pressing it directly on the surface of the pastry cream. Cool to room temperature, about 2 hours.
Using an electric mixer, beat the unsalted butter in a large bowl until fluffy. Add the pastry cream ¼ cup at a time, beating well after each addition. (The frosting can be prepared up to 2 days ahead, simply cover it and refrigerate. Bring to room temperature before using.)