Cook pasta per package directions. Reserve 1 cop of pasta water before draining.
In food processor, pulse cheese, pistachios, mint, parsley, and chives while drizzling in oil and 1/2 cup pasta water in a steady stream. If necessary, slowly add some of the remaining pasta water until the pesto is fluid but not runny.
Transfer pesto, pasta, and chicken to a bowl and toss to coat.
Serve with additional cheese, pistachios, and chives.