Preheat the oven to 400ºF. Lightly grease a 12-cup muffin tin.
In a large bowl, stir together the oil, vinegar, mustard, herbs, garlic, Parmesan, and a generous pinch of salt and pepper. Add in the potatoes and toss to coat.
Place the potatoes in the muffin cups, dividing evenly and stacking them to the top. Cover the muffin tin loosely with foil and place on a baking sheet. Bake for 30 minutes. Remove foil and continue baking until the potatoes are tender and golden, 20-25 minutes more. Gently run a butter knife around the edges of each cup to release the stacks from the pan. Garnish with more parmesan and salt. Serve immediately.