Scrub the potatoes clean, if necessary, then submerge them in a pan of lightly salted water. Bring to a boil and simmer until tender.
Meanwhile, grate the cheeses. Once the potatoes are tender, drain them and use a tea towel to rub off their skins (obviously they will be very hot).
Put them through a food mill or potato ricer, then push them through a sieve or mash them very well, until you have a smooth puree. Crush the garlic clove.
Put the puree back into the pan and put on a very low heat. Stir in the creme fraiche and garlic along with enough milk to make a viscous puree, then use a wooden spoon or sturdy whisk to stir in the cheese in a couple of batches.
Beat the potato vigorously until the cheese is melted (and the texture smooth) and the mix comes away from the side of the pan. It will develop a slight sheen that lets you know it’s ready – add more milk if it seems too firm.
Season with salt and white pepper and serve immediately.