Mix together the soy sauce, ginger, and mirin in a large bowl to make the marinade. Reserve 1/2 cup of the marinade and set aside. Lay the squid in the remaining liquid, gently turning 4 times to coat all over. Marinate the squid for 10 minutes at room temperature, turning once.
Preheat a grill to hot. Grill the squid for about 4 minutes (5 minutes if they're larger than 1 ounce apiece). Every 30 seconds flip the squid and brush on the reserved marinade. The squid will turn from translucent to white when they're ready, becoming tender and releasing a rich aroma. Be careful not to overcook; squid turns rubbery if grilled too long. Serve immediately.
