In a small saucepan over medium heat, melt butter. Brush 12 muffin cups generously with butter. Place 1 thyme sprig in center of each muffin cup.
Add chopped thyme and garlic to remaining butter in saucepan and stir over medium heat until fragrant, about 2 minutes. Remove from heat and set aside.
Using a mandoline, slice potatoes crosswise into very thin rounds (1/16"-ish). Place in a large bowl. Pour herb butter over potatoes and season with salt and pepper. Toss well to coat.
Divide potato slices among muffin cups, layering overlapping slices to create a circular pattern. When the muffin cup is filled to the rim, press down lightly with a fork to compact. Drizzle any remaining butter in bowl over potatoes.
Cover muffin tins tightly with aluminum foil. Place muffin tin(on a baking sheet and bake in preheated oven until potatoes can be easily pierced with a knife, about 35 minutes. Remove foil.
Prepare another baking sheet by lining with parchment paper. Invert muffin tins onto prepared sheet, banging it a bit to loosen potatoes. If any potatoes/thyme has stuck to the muffin tins, gently use a butter knife to loosen and place on potato stacks. [IF MAKING AHEAD, STOP HERAllow potatoes to cool a bit, then cover baking sheet with plastic wrap and refrigerate. Proceed with rest of recipe when ready.]
Increase oven temperature to 425° [If you made ahead, remove baking sheet with potatoes, remove plastic wrap and preheat oven to 425° F., allowing potatoes to sit at room temperature while oven heats]. Place into 425° oven and bake for 10 minutes. Remove from oven and carefully using a spatula, flip potato stacks over (thyme side dowand bake for another 8-10 minutes, until under-side is nice and golden. Remove from oven and flip thyme-side-up to serve. [If you made ahead and your potatoes were cold/room temperature when they went in to the oven, you may need a few minutes longer on each side.]