Oven Preheating: Initiate by setting the oven temperature to 350°F (180°C), suitable for all oven types.

Butter Application to Muffin Tin: Melt the butter over medium heat in a saucepan. Utilize a portion to lightly coat the interior of the muffin tin holes. Reserve the remaining butter for later use.

Preparation of Cream Sauce: Add minced garlic to the remaining butter in the saucepan, cooking for approximately 20 seconds. Follow this by introducing the cream and salt, bringing the mixture to a simmer. Allow simmering for 30 seconds, then remove from heat and keep warm.

Potato Slicing: Begin by peeling the potatoes. Cut their bases to create a flat surface so they stand upright. Fashion the potatoes into cylinder shapes that fit within the muffin tin. Proceed to slice these cylinders into 2mm thick pieces, utilizing either a mandolin or a sharp knife for precision.

Assembly of Potato Stacks: Insert potato slices into the muffin tin, arranging them until they reach halfway up the sides of the tin holes. Strive for uniformity in size to form neat stacks.

Application of Cream Mixture: Equitably distribute approximately 1 teaspoon of the cream mixture onto each potato stack, utilizing half of the total mixture prepared.

Cheese Layering: Place a single square of cheese atop each potato stack.

Final Layering: Add the remaining potato slices atop each stack, ensuring the height extends about 1cm above the muffin tin rim, as they will settle once baked. Drizzle the remaining cream mixture and garnish with most of the thyme, reserving a small amount for final garnishing.

Initial Baking Phase: Cover the muffin tin loosely with foil and bake for 40 minutes. The potatoes should be tender enough to be pierced easily with a small sharp knife.

Final Baking Phase with Cheese: Following the initial bake, remove the tin from the oven. Sprinkle the shredded cheese over the potato stacks and return to the oven, uncovered, for an additional 10 minutes until the cheese is golden and bubbly.

Final Garnish and Resting: Once baked, embellish the gratins with the remaining thyme. Allow the gratins to rest for 5 minutes before serving. A tablespoon or butter knife can be employed to assist in removing them from the tin.

Serving Suggestions: These gratins can be served as a sophisticated side dish with dinner, as a standalone finger food, or as part of a hearty breakfast accompanied by eggs and bacon.