Bring a medium pot of salted water to a boil. Blanch basil leaves for 10 seconds, then remove with a spider or slotted spoon and transfer to a bowl of ice water. Drain thoroughly.

In a food processor or mini chopper, pulse pistachios 5-7 times until roughly chopped. Add blanched basil and mint leaves and pulse a few times more. Add garlic and pulse to combine.

With the machine running, slowly drizzle in olive oil until fully incorporated.

Turn off the processor, add Parmigiano Reggiano, and pulse a few more times to mix.

Add pepper and kosher salt to taste, then pulse again briefly to combine.