Preheat the oven to 180C/350F/Gas 4.
For the sautéed mushrooms, heat half of the olive oil with the butter in a frying pan and lightly fry the mushrooms and garlic for five minutes. Squeeze over the lemon juice, drizzle over the balsamic vinegar and cook for a further 4-5 minutes, or until reduced slightly. Stir in the remaining olive oil and the chopped chives.
For the monkfish, heat the olive oil and butter in an ovenproof frying pan. Season the fish with salt and freshly ground black pepper, then add to the pan and fry for 2-3 minutes on each side, or until golden-brown. Squeeze over the lemon juice, then transfer to the oven to roast for 6-8 minutes, or until the fish is cooked through. Remove from the oven and rest for five minutes, then thickly slice.
For the 'mushroom' tatties, heat the olive oil in an ovenproof frying pan and lightly fry the mushroom-shaped potatoes for 4-5 minutes, or until they start to colour. Add the garlic and rosemary, then place the pan in the oven to roast for ten minutes, or until the potatoes are golden-brown and tender.
To serve, pile the sautéed mushrooms onto a serving plate and place the sliced monkfish on top. Spoon the roasted 'mushroom' tatties around the edges of the plate and serve.