How to Make It
Preheat the oven to 350°. Scoop up 1/3 cup of the mashed potatoes, press a slice of the cheese in the center and pat the mashed potatoes around the cheese to enclose it. Flatten the potato cake slightly. Repeat with the remaining mashed potatoes and cheese.
In a medium cast-iron skillet, melt the butter in the oil over high heat. When the fat begins to brown, carefully scoop two 1/2-cup mounds of shredded potato into the skillet; flatten them with a spatula and season with salt and pepper. Top each with a mashed potato round and then cover with another 1/2 cup of the shredded potatoes. Season with salt and pepper, reduce the heat to low and cook, turning once, until the cakes are crisp and golden, about 8 minutes per side. Transfer the potato cakes to a baking sheet and keep warm in the oven. Repeat to make the remaining 2 potato cakes. Sprinkle the potato cakes with the thyme and serve hot.