Add peeled and cubed potatoes into a medium pot and cover with cold water 2 inches above the potatoes. Add in the garlic and 1 tbsp salt, then bring it to the stove over medium heat. Bring up to a boil, then drop to a simmer until a knife easily pierces potatoes with no resistance, about 20 minutes. Use a mug or measuring cup to remove some cooking liquid right before draining the potatoes in a colander, shaking well to get rid of excess water.

While the potato cooks, add the heavy cream, lemon peels and fresh thyme into a small pot over a simmer. Grate the cheeses and combine them together.

Use a potato ricer or food mill to puree the potatoes back into the same dried-out pot. Add in the butter and using a silicone spatula, stir potatoes well until butter is fully combined. Then slowly add the steeped cream in 3 parts (removing the lemon and thyme first). Stir well until cream is fully incorporated before the next addition. Add the nutmeg to taste.

Add the grated cheese in two batches, stirring well 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). The aligot should form long, stretchy strands when you lift it from the pot.

Season with salt and serve immediately.