Preheat oven to 350°F. Melt butter in a small saucepan over medium heat. Brush muffin cups all over with butter. Line bottoms with parchment-paper rounds. Arrange 1-2 small thyme sprigs in center of each round. Drizzle 1/2 teaspoon butter into bottom of each cup.

Add chopped thyme and garlic to remaining butter in saucepan. Stir over medium-low heat until fragrant, about 2 minutes. Remove from heat.

Using mandoline, slice potatoes crosswise into very thin rounds (less than 1/16″ thick), placing them in a large bowl as you work. Pour herb butter over and season with salt and pepper; toss to coat well.

Divide potato slices among muffin cups, layering overlapping slices to create a circular pattern. Lightly press center of each to make compact. Drizzle any remaining butter and seasoning from bowl over.

Cover muffin pan tightly with foil and place on a baking sheet. Bake until potatoes can be pierced easily with the tip of a knife, about 35 minutes. Remove foil; invert a rimmed baking sheet over pan. Turn, lightly tapping on counter, releasing potatoes onto sheet. Rearrange any slices that may have fallen out. Using a metal spatula, carefully turn cakes, thyme sprigs facing down. Discard parchment. DO AHEAD:Potatoes can be made 1 day ahead. Cover; chill.

Increase heat to 425°F. Uncover cakes if needed. Bake until bottoms and edges are golden and crispy, 25-30 minutes. Carefully turn cakes, thyme sprigs facing up. probably not necessary to turn them