Preheat the oven to 180°C/350°F/gas mark 4. Use the butter to grease a 25 cm (10 in) gratin dish. Arrange the turnips overlapping in layers in the dish, interspersed with the oven-dried tomatoes and seasoning between the layers with just a little salt – the tomatoes can be quite salty. Arrange the final layer neatly, using both turnips and tomatoes.
Bring the cream, milk, garlic and thyme to the boil, then remove from the heat and stir in the mascarpone. Pour through a sieve on to the turnips, ensuring the liquid covers them. Sprinkle with the cheese and bake for 40-45 minutes or until the turnips are tender and the top is golden and crusty.