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In a large saucepan, cover potatoes and 1 whole clove garlic with cold water by at least 2 inches. Add thyme. Season water with salt until water is salty like tears. Bring water to a boil over high heat, then reduce heat to medium-low and simmer until a knife easily pierces potatoes with no resistance, about 20 minutes. Drain potatoes in a colander; discard thyme.
Using a ricer or a food mill fitted with the finest disk, purée potatoes back into saucepan, mixing butter cubes in as you go. (If using a food mill, you can add the butter to the potato chunks and work it through the mill; if using a ricer, add the butter to the saucepan with the riced potatoes.) Add remaining 1 clove minced garlic. Set saucepan over medium-low heat. Using a silicone spatula, stir potatoes well until butter and garlic are fully combined. (Alternatively, you can use a wooden spoon for this, though it won't conform to the sides of the pot like silicone will.)
Add cream and stir well until cream is fully incorporated. Continue stirring and working potatoes, lowering heat to low if potatoes begin to sizzle and steam, until potato mass feels thickened and sticky, about 3 minutes.
Add grated cheese in small batches, stirring between each addition until cheese is fully melted and incorporated. Continue stirring potatoes until they become thick, silky, smooth, and elastic (do not worry about overworking the potatoes in this recipe, since you want to develop that starch), about 3 minutes longer. The aligot should form long, stretchy strands when you lift it from the pot.
Season with salt. The aligot should be thick, with a very slow-flowing viscosity, like cooling lava; if too thick, work in additional cream in small amounts until it is loose enough.
Transfer to a warmed serving dish and serve right away. Aligot is traditionally served with sausages and meats, but you can also serve it with roasted vegetables or on a bed of polenta. Leftover aligot can be refrigerated in a sealed container. To reheat, add to a saucepan with a small amount of cream. Bring cream to a simmer before stirring it in, then continue to cook, adding more cream as necessary, until aligot is heated through and consistency is correct.