Heat the oil and butter on a griddle over medium-high heat until the butter is melted. Add the shredded potatoes and pack them down tightly into a patty. Cover and cook just until the perimeters start to get golden, about 5 minutes. Use a spatula to press the top down. Season with salt and pepper and flip. Cook, uncovered, until the bottom is golden and crispy, another 3 to 5 minutes.