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Bake the gratin up to 2 days ahead; leave to cool, then cover with a sheet of baking paper and chill. To reheat, bake at 180°C, fan 160°C, gas 4, for 30-40 minutes.
Preheat the oven to 160°C, fan 140°C, gas 3. Butter a 7.5cm-deep 24 x 30cm ovenproof baking dish (or 12 x 300ml individual dishes). Halve one of the garlic cloves and rub the inside of the dish(es).
Lightly squash the remaining garlic cloves with the flat of a kitchen knife, then place all three cloves in a large pan with the cream, milk, crème fraîche and bay leaves. Heat on the hob until just below boiling point, add the nutmeg and season well. Set aside while you prepare the potatoes.
Cut both types of potato into slices about the thickness of a pound coin – if you have an attachment on your food processor use this, as it’s much quicker.
Arrange a third of the Desiree potatoes in an even layer in the base of the baking dish, or divide among the individual dishes, then add half the sweet potatoes, seasoning as you go. Repeat with further layers of white, sweet, then white potato. Strain the cream mixture over the potatoes to just cover the top layer.
Cover with foil and bake in the oven for 1 hour 45 minutes until the potatoes are almost tender. Remove the foil, scatter the Parmesan on top and cook for a further 15 minutes until bubbling and golden. Leave to rest for 5-10 minutes before serving.