Start by preheating your oven to 350°F (180°C). Then, melt a bit of butter in a small saucepan. Use a pastry brush or paper towel to lightly coat the inside of your 12-cup muffin tin holes. Keep the rest of the melted butter for our delicious cream sauce!
In the same saucepan with the leftover melted butter, add your minced garlic and cook for about 20 seconds until fragrant. Pour in the heavy cream and a sprinkle of salt. Bring it to a gentle simmer for just 30 seconds to let the flavors mingle, then take it off the heat and set aside.
Time to prep the potatoes! Peel your potatoes and carefully trim the ends so they can stand flat. Then, shape each potato into a cylinder that fits snugly into your muffin tin cups (don't worry if they're not perfectly round!). Use a mandoline or a very sharp knife to slice them super thin, about 1/10th of an inch (2mm).
Now for the fun part: assembly! Stack the thin potato slices into each muffin tin cup until they reach about halfway up. Try to keep the stacks neat and uniform for beautiful results.
Drizzle about 1 teaspoon of your creamy garlic sauce over each potato stack, using roughly half of the mixture you prepared.
Place one of your thin cheese slices directly on top of the cream-drizzled potatoes.
Add another layer of potato slices on top of the cheese, building each stack so it's slightly above the muffin tin rim (they'll settle as they bake). Pour the remaining cream mixture evenly over each stack and sprinkle most of the fresh thyme leaves over them, saving a little for garnish.
Loosely cover the muffin tin with aluminum foil. Bake for 40 minutes, or until the potatoes are perfectly tender when pierced with a small knife – it should slide through easily without resistance.
Carefully remove the foil. Sprinkle the shredded Gruyere cheese (or your chosen cheese) over the top of each stack. Return to the oven, uncovered, for another 10 minutes, or until the cheese is beautifully melted and golden brown.
To finish, sprinkle with the reserved fresh thyme. Let them cool in the muffin tin for about 5 minutes before gently removing them with a spoon or butter knife. Serve warm as a delightful side dish, impressive party appetizer, or even for a fancy breakfast!