Preheat the oven to 425°F (220°C). Line a baking sheet with parchment paper.

Place each potato between two chopsticks or wooden spoons. Slice thin vertical cuts across the top, stopping when the knife hits the chopsticks so the potato stays intact.

Toss the potatoes with olive oil, salt, pepper, and thyme until evenly coated.

Spread the shredded Parmesan into thin piles on the baking sheet, roughly matching the number of potatoes.

Place the potatoes cut-side up directly on top of the Parmesan.

Roast for 30 to 35 minutes, until the potatoes are tender and the Parmesan is golden and crisp around the edges.

While the potatoes roast, make the pesto. Blend basil, garlic, and Parmesan until finely chopped. With the motor running, stream in olive oil until smooth. Season with lemon juice and salt.

In a small bowl, gently swirl the pesto into the Greek yogurt. Do not overmix. Add a pinch of salt.

Let the potatoes cool slightly, then transfer to a serving plate and serve with the pesto yogurt dip.