On a work surface, remove the tip of calamari. With a knife, open into large squares. With a sharp knife, score the flesh of each calamari square in a crisscross pattern, without piercing the flesh. Place in a large sealable bag. Add 30 ml (2 tablespoons) of oil, the lime juice, and garlic and marinate for about 15 minutes.
Preheat the grill, setting the burners on high. Oil the grate.
Remove the calamari from the marinade. Season with salt and pepper. Grill for about 30 seconds on each side.
In a bowl, combine the lettuce with the remaining olive oil. Season with salt and pepper. Serve the warm calamari over the salad drizzled with spicy mayonnaise.