Preheat the oven to 350°F and lightly grease a 1-quart casserole dish.
Melt the butter in a small saucepan over medium-low heat. Whisk in the flour until smooth, then cook for about 1 minute.
Slowly add the cold milk while whisking continuously to keep the mixture smooth.
Season with salt and a dash of cayenne pepper. Continue cooking over low heat, whisking occasionally, until the sauce thickens and begins to gently bubble.
Reduce the heat and stir in 1 cup of sharp cheddar cheese. Continue stirring until the cheese has completely melted and the sauce is creamy. Remove from the heat.
Arrange half of the sliced potatoes in an even layer in the prepared casserole dish.
Pour half of the cheese sauce evenly over the potatoes.
Add the remaining potatoes, then cover them with the rest of the cheese sauce.
Sprinkle the remaining 1/2 cup of grated cheese over the top and finish with a light dusting of paprika.
Bake uncovered for about 1 hour, or until the potatoes are tender when pierced with a knife and the top is bubbling and golden.
Remove the casserole from the oven and let it rest for about 10 minutes before serving.