Preheat the oven to 425°F and line a baking sheet with parchment paper. Place each potato between two chopsticks or wooden spoon handles and slice crosswise every 1/8 inch, letting the chopsticks stop you so the bottom stays connected.
Stir together the melted butter and olive oil, then pour half of it into a separate small bowl and set aside. Brush the potatoes all over with the first half, season with salt and pepper, and arrange them on the baking sheet.Roast for 30 minutes, until the slices begin to fan open.
Stir the garlic, thyme, and rosemary into the reserved butter mixture. Brush it generously over the potatoes, nudging it down between the slices.
Return the potatoes to the oven and roast for another 25-30 minutes, brushing once more with the pan juices halfway through, until the edges are crisp and the centers are tender.
Sprinkle with flaky sea salt and fresh parsley and serve hot.