Those irresistible crisp, lacy edges are the best part of scalloped potatoes, so we make them in a sheet pan—increasing the surface area for maximum browning. The sauce is packed with plenty of sharp Cheddar that goes perfectly with thin slices of tender potato.
Preheat the oven to 450 degrees F. Lightly grease an 18-by-13-inch sheet pan with the butter.
Bring the cream, thyme, garlic, 1/2 teaspoon salt and 1/4 teaspoon pepper to a simmer in a small saucepan over medium heat, then turn off the stove.
Thinly slice the potatoes on a mandoline or vegetable slicer into rounds about 1/8 inch thick. Arrange half the potato slices in a single, overlapping layer on the sheet pan and sprinkle evenly with salt, half the Cheddar and half the Parmesan. Top with the remaining potato slices and sprinkle with salt. Drizzle the hot cream mixture evenly over the potatoes. Use a small offset spatula or butter knife to spread the garlic and thyme evenly over the potatoes.
Cover the pan with foil and bake until the potatoes are tender when pierced with a paring knife, about 20 minutes. Carefully remove the foil and sprinkle the potatoes with the remaining Cheddar and Parmesan. Cook uncovered, rotating the pan halfway through, until very brown and crusty on top, 15 to 20 minutes more.