Cook the peeled potatoes in boiling water until quite tender (about 20 to 25 minutes).
Meanwhile in a small saucepan warm the cream, butter, and garlic until almost simmering.
Drain the potatoes and either smash them with a potato masher or run them through a stainless steel food mill.
Place a solid pot (ideally a Dutch oven) on low heat and place the potatoes in this. Stir in the warmed cream mixture and a pinch of pepper (you might not need any salt as the cheese will add quite a bit). Using a wooden spoon, begin gradually and vigorously stirring in the cheese bit by bit.
Traditional aligot recipes tell you to form figure eights while you are doing this, but circles work fine as well. If you occasionally pull the spoon up out of the pot, you should see more and more ribboning as the cheese gradually melts.
Continue beating until the potatoes come away from the sides of the pan and you get long smooth ribbons when you lift the spoon through the potatoes. The whole process of incorporating the cheese will take about between 10 and 15 minutes. Season to taste with salt and pepper. Serve immediately.