1.To make aioli, using a mortar and pestle, pound garlic and salt to a paste. Add anchovies, pound until smooth. Add egg yolk then whisk in oils, a few drops at a time, until combined. Stir in juice; season to taste. (Or, blend garlic, salt, anchovies and egg yolk until smooth. Gradually add oil in a thin stream, until thickened and combined. Add juice; season.)
2.Cook potatoes in a saucepan of boiling salted water for 2 minutes to par-cook; drain immediately and spread out on a large tray to dry.
3.To clean calamari, gently pull the head and entrails away from the body, then remove and discard the clear quill. Cut the tentacles from the head just below the eyes, then remove the beak. Gently pull away the outer membrane from the hood. Cut tentacles in half. Cut hood lengthways to open out; score the inside finely in a criss-cross pattern, without cutting all the way through. Cut calamari into 4cm x 6cm (1½-inch x 2½-inch) pieces.
4.Heat a barbecue plate (or chargrill pan) on medium heat. Toss potatoes in paprika and half the oil; season well. Barbecue potatoes for 7 minutes each side or until browned and tender. Transfer to a medium heatproof bowl; cover to keep warm.
5.Increase heat to high. Toss calamari in remaining oil; season. Barbecue calamari, smooth-side down first, for 2 minutes each side or until curled, charred lightly and just tender.
6.Combine rocket, potatoes and calamari on a large platter. Drizzle with some of the aioli. Serve with the remaining aioli and lemon cheeks.
This recipe would serve 4 as a main course with bread. You can ask the fishmonger to clean the calamari for you. Cleaned squid hoods are available from fish shops, but are usually too large and thick. Any leftover aioli is delicious on sandwiches instead of mayonnaise or served with grilled seafood or chicken.