Perfect for brunch or dinner side dish, these potatoes are both beautiful and tasty. And you can do the first bake ahead then refrigerate and do the final bake when you're ready.
Preheat oven to 350° F. (regular bake setting/not fan assisted)
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) Allow 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°F oven and bake for 10 minutes. Remove from oven and carefully using a spatula, flip potato stacks over (thyme side down) and 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.)