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.