Peel the turnips and cut them into quarters. Rinse them with the broccoli and cauliflower florets. Rinse the fish fillets and pat them dry.
Spread the turnips in the perforated part of a steamer. Place them above the boiling water and cook for 5 minutes before adding the broccoli and cauliflower. Leave to cook for another 8 minutes: prick the vegetables with the tip of a knife, they should be just tender.
Set the vegetables aside in a salad bowl and put the fish fillets in their place. Cook for 4 to 5 minutes depending on the thickness of the fillets.
Coarsely chop the nuts. Rinse the chives and pat them dry. Mix with a fork the pressed lime oil, the pomegranate balsamic and the green fruity extra virgin olive oil.
Divide the fish and vegetables between four plates and drizzle with the vinaigrette. Sprinkle with walnuts and chives and serve immediately.