To clean the artichokes, cut off the stems flush with the base, remove any tough, unsightly outer leaves and cut off the top one-third of the flower bud itself. Force the leaves open a little, and rinse under cold water, with a squeeze of lemon or a splash of vinegar to prevent discoloring.
Place the artichokes upside-down in a pot large enough to hold them all in one layer and pour in boiling water to come one-fourth of the way up the sides. Salt and simmer 15-40 minutes, depending on their age and size - until the heart is just tender when pricked with a fork. Drain and refresh them under cold water; drain again and chill.
Beat the tarama or smoked cod's roe with the grated onion. You can do this easily in a food processor. Finely grate the zest of one lemon and beat in, with 2 teaspoons boiling water.
Squeeze the juice of both lemons. Alternately dribble in lemon juice and olive oil, beating each addition in before adding the next, as though making mayonnaise - it may not need all the oil. If you are doing this by hand, it take perseverence and elbow grease to make it nice and light. Taste and season with pepper. Open the artichokes and remove the inner, soft leaves and then the hairy choke. Spoon tarama into the cup of leaves.