Trim and discard root ends of green garlic. Finely chop green garlic, rinse thoroughly and spin dry.
In a large frying pan over medium-high heat, cook olive oil, green garlic, and 1/2 tsp. salt until soft, about 3 minutes. Let cool to warm room temperature.
In a blender or food processor, pulse pistachios to chop. Set aside. Add green garlic & basil and process, scraping down sides as necessary, until bright green and smooth. With motor running, drizzle in olive oil. Pulse in reserved nuts and cheese. Taste and add more salt if you like.
Gnocchi
Combine all the gnocchi ingredients in a mixer with the paddle attachment and beat on low till combined fully.
Check mixture, it should be quite wet and sticky, add more flour if needed.
Turn mixture onto floured work surface, it should resist coming out of the bowl
Using only as much flour as necessary, role dough into dowels about the circumference of a nickel.
Cut dowel into pieces about the length of your thumbnail, and place on a sheet tray making sure they do not touch.
Freeze gnocchi till ready to use, once frozen, gnocchi can be portioned into plastic bags.
Boil gnocchi in salted boiling water until they float.
To Serve
Once your gnocchi is cooked, strain and place in a bowl and add enough pesto to generously coat the gnocchi.
Drizzle with extra virgin olive oil and a handful full of pecorino cheese.