Clean, wash, peel and quarter the turnip and cut into thin slices (approx. 3 mm). Clean, wash, peel and slice the carrots. Mix milk, cream and stock, season with salt, pepper and nutmeg.
Place the turnip and carrots in fan-shaped layers in an ovenproof dish, pour the prepared liquid over them and bake in a preheated oven (electric oven: 175°C/ gas: level 2) for about 1 hour.
Wash the thyme, dab dry and pluck the leaves from the stems. Sprinkle gratin with cheese and thyme halfway through the baking time and bake until done. Wash the meat, dab dry, remove from the bone and rub with salt and pepper.
Heat the oil in a pan and fry the chicken breast on both sides for about 5 minutes at medium heat. Cut the meat into slices. Arrange gratin with the chicken breast on plates. Serve garnished with parsley and thyme.