Toss the potatoes with 2 tablespoons of the oil and salt and pepper in a small, sturdy roasting pan. Roast in a preheated oven, 200°C (400°F), Gas Mark 6, for 15 minutes.
Mix together the breadcrumbs, garlic, fresh coriander, spices and seasoning. Cut away any thick areas of fat from the skinned side of the lamb. Brush the lamb with the egg yolk and spoon the breadcrumb mixture over, pressing down gently with the back of the spoon. Brush the mushrooms with the remaining oil and a little seasoning.
Turn the potatoes in the pan and add the lamb, crusted-side uppermost. Return to the oven for 30 minutes. (The cutlets will still be slightly pink in the middle after this time, so cook for a little longer if you prefer them well done.) After 15 minutes of the cooking time, turn the potatoes in the oil and add the mushrooms. Return to the oven for the remaining cooking time.
Drain the meat to a board. Cover with foil and leave to rest for 15 minutes. Transfer the potatoes and mushrooms to a warmed serving dish.
Add the broad beans, mint and wine to the roasting pan and cook over a gentle heat for 5 minutes until the beans are tender. Tip into a blender or food processor and blend until smooth. Check the seasoning and spoon on to warmed serving plates. Carve the lamb into cutlets and add to the plates with the mushrooms and potatoes.