Set the oven to 400°F and generously grease a 12-cup muffin tin. A continuous coating across the bottoms and sides helps the browned cheese release later.

Prepare uniform 1/8-inch potato rounds, leaving the sliced surfaces unrinsed. The rounds should feel slightly starchy rather than washed clean, which helps the cooked layers hold together.

Whisk the melted butter, pressed or grated garlic, sea salt, and cracked pepper in a large bowl, then turn the potato slices through the mixture. Separate any rounds stuck together so each surface looks lightly glossy with seasoned butter.

Scatter ½ cup of the Parmesan and the chopped thyme over the coated potatoes, then gently redistribute the slices. Small flecks of cheese and herbs should reach throughout the bowl rather than collecting on the uppermost rounds.

Build the stacks inside the prepared cups up to the top rim, distribute the butter left in the bowl among them, and finish with the remaining ¼ cup Parmesan. Keep the layers level and supported by the cups, allowing for some settling as the potatoes roast.

Bake for 45 to 50 minutes, checking the center of the thickest stack with a paring knife near the end. Look for sizzling butter, golden edges, and a lacy cheese skirt, with no firm resistance when the knife enters the center.

Leave the baked stacks in their cups for 5 minutes before easing them out with a thin spatula or spoon. The pause lets the potatoes absorb some pooled butter and helps the tender layers stay together.

Arrange the stacks on a platter and finish with flaky sea salt, additional cracked pepper, and the thyme leaves. The fresh garnish adds herbal aroma against the warm, browned cheese.