Preheat oven to 425°.
In a large pot, cover potatoes with water and add a generous pinch of salt. Bring water to a boil and simmer until potatoes are tender, about 15 minutes. Drain and let sit until cool enough to handle.
On a large rimmed baking sheet, toss potatoes with melted butter, garlic, and thyme. Using the bottom of a small glass or mason jar, press down on potatoes to smash them into flat patties.
Season with salt and pepper, then sprinkle with Parmesan.
Bake until the bottoms of potatoes are beginning to crisp and Parmesan is golden, about 25 minutes.