Toss the potato slices with 4 tablespoons of the oil, a little salt and pepper and the thyme in a bowl. Tip into a roasting tin or ovenproof dish and spread out in an even layer. Cover with foil and bake in a preheated oven, 190°C (375°F), Gas Mark 5, for about 30 minutes until the potatoes are tender.
Score the bream fillets with a sharp knife. Mix the prosciutto with the shallots, lemon rind and a little pepper and use to sandwich the bream fillets together. Tie at intervals with string. Cut each of the sandwiched fillets through the centre to make 4 even-sized portions.
Lay the fish over the potatoes and return to the oven, uncovered, for a further 20 minutes or until the fish is cooked through.
Scatter the samphire around the fish and drizzle with the remaining oil. Return to the oven for a further 5 minutes before serving.