Preheat oven to 350 degrees Fahrenheit.

Brush 12 muffin cups or ramekins all over with oil. Line bottoms with parchment-paper rounds. Drizzle 1/2 teaspoon oil into bottom of each cup.

Heat remaining oil in a frying pan over medium heat. Add diced onion and stir over medium-low heat until translucent, about five minutes. Remove from heat.

Using mandoline, slice potatoes crosswise into very thin rounds (less than 1/16-inch thick), placing them in a large bowl as you work. Pour sautéed onion 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 oil and seasoning in the bowl over the potatoes.

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 potato stacks over. Remove parchment. (Potatoes can be made one day ahead up to this point; cover and refrigerate.)

Increase heat to 425 degrees Fahrenheit. Uncover potato cakes. Bake until bottoms and edges are golden and crispy, 25–30 minutes.