Preheat the grill to medium.
For the crusted lamb steak, heat the oil in an ovenproof frying pan. Season the lamb steak with salt and freshly ground black pepper and fry for three minutes on each side.
Mix the cheddar and breadcrumbs together in a small bowl and season with salt and freshly ground black pepper.
Press the cheese mixture on top of the steak. Place under the grill and cook for 2-3 minutes, or until golden-brown and bubbling.
Meanwhile, for the carrot purée, place the cooked carrot into a food processor with the butter and oil. Season, to taste, with salt and freshly ground black pepper and blend until smooth.
To serve, place the carrot purée onto a serving plate and top with the lamb.