Add fish to a food processor and pulse until finely chopped. Add 1 tablespoon ice water and continue to pulse to a paste. Add the cold butter, mirin, sesame oil, ginger, garlic, soy sauce, salt, and pepper. Pulse to combine thoroughly. Scrape down the bowl and continue to pulse until the mixture is very smooth and the butter is very fine. Transfer to a bowl and fold in garlic chives. Cover and refrigerate 1 hour or overnight for the flavors to come together.