Preheat your oven to 425°F (220°C). Line a baking dish or sheet pan with parchment paper.
Place one potato on a cutting board between two wooden chopsticks or spoon handles. Slice crosswise at ⅛-inch (3mm) intervals, letting the chopsticks stop your knife before cutting through. Repeat with all potatoes.
In a small bowl, whisk together the melted butter and olive oil. Season with salt and pepper.
Place the potatoes cut-side up in the prepared baking dish. Brush generously with half the butter-oil mixture, working it down into the slices.
Roast for 30 minutes. Remove from the oven and brush again with the remaining butter-oil mixture. Use a fork or your fingers to gently fan the slices open a little further.
Tuck thin garlic slices and small thyme sprigs down between the potato fans. Return to the oven.
Roast for another 20–25 minutes, until the edges are deeply golden and crispy and the centers are tender when pierced with a knife.
If using cheese, scatter it over the potatoes in the final 10–15 minutes of cooking.
Remove from oven and rest for 5 minutes. Finish with an extra pinch of flaky sea salt and serve immediately.
