Rinse potatoes in a colander under cold water until water runs clear. Drain well and squeeze firmly between kitchen towel to remove excess moisture. Transfer to a mixing bowl and toss with salt, pepper, and paprika.

Heat one tablespoon of butter or oil in a large oven-safe skillet over medium-high heat. Add potatoes and press gently into an even layer. Cook until a golden brown crust forms on the bottom, about 5 minutes. Slide hash brown onto a plate and coat the bottom of skillet with the remaining tablespoon of butter or oil. Carefully return the hash brown to the skillet, browned side up, and cook until bottom is golden brown, about 5 minutes longer.

Top with shredded cheese and place under the broiler until the cheese is melted and bubbling.

Garnish with scallions or parsley. Cut into wedges and serve.