Clean, wash and thinly slice the potatoes and carrots. Bring the stock to the boil, add the carrots and potatoes and simmer for about 5 minutes. In the meantime wash fish fillet, dab dry and cut into 4 pieces.
Sprinkle fish slices with lemon juice, salt lightly and place in an ovenproof dish. Wash the dill, dab dry and chop except for something to garnish. Stir half of the dill into the mustard and brush the fish fillets with it.
Pour the carrots and potatoes onto a sieve and collect the broth. Let the potatoes and carrots cool down a little, place them in flakes on the fish slices, season with pepper and sprinkle with parmesan.
Spread the rest of the vegetables around the fish and pour on approx. 1/8 litre of vegetable stock. Bake in the preheated oven (electric cooker: 225 °C/ convection oven: 200 °C/ gas: level 4) for 15 minutes. Clean, wash and drain the lamb's lettuce.
Mix vinegar with salt and pepper and fold in the oil. Turn the lamb's lettuce in the vinaigrette. Sprinkle the finished fish fillets with the remaining dill and garnish. Arrange on plates with lamb's lettuce and serve immediately.