Preheat the oven to 425 degrees F. Line a baking sheet with aluminum foil.
Make a series of 1/8-inch slices along with each potato, slicing 2/3 of the way through.
Stir together the butter, oil, thyme, garlic, 1/4 teaspoon salt and 1/4 teaspoon pepper in a small bowl. Rub the potatoes all over with the mixture, getting in between the slices.
Place on the baking sheet and roast until the center of the potatoes are tender and the outside is crisp, 50 minutes to 1 hour. Halfway through the roasting time, remove the potatoes from the oven and run a fork gently across the tops of the potatoes, using light pressure, to fan the slices and separate them from one another.
Meanwhile, stir the yogurt and scallions with a pinch salt and pinch pepper. Serve the sauce with the potatoes.