Preheat Oven & Prep Dish: Preheat your oven to 350°F (175°C). Lightly grease a one-quart casserole dish.
Make the Cheese Sauce: In a small saucepan, melt the butter over low heat. Whisk in the flour and cook for about one minute. Gradually add the cold milk, stirring continuously with a whisk until the sauce is smooth and lump-free. Season with salt and a dash of cayenne pepper.
Cook the Sauce & Add Cheese: Increase the heat to medium and continue to cook, stirring with a whisk until the sauce comes to a boil and thickens. Reduce the heat to low and stir in 1 cup of the grated sharp Cheddar cheese until it is completely melted and smooth.
Layer & Bake: Place half of the sliced potatoes in the bottom of the prepared casserole dish. Pour half of the cheese sauce evenly over the potatoes. Repeat with the second layer of potatoes and the remaining cheese sauce.
Garnish & Finish: Sprinkle the remaining ½ cup of grated cheese on top of the casserole. For a pop of color, you can also sprinkle a little paprika on top.
Bake uncovered for about 1 hour, or until the potatoes are tender and the top is golden and bubbly. Serve hot.