Preheat oven to 375°F. Grease a casserole dish. Wash and cut potatoes in 1/8 inch rounds.
In a medium saucepan, over medium-low heat melt butter. Slowly whisk in flour. Continue whisking until flour is a light brown. Slowly start to add milk and half n half alternating between the two. Add salt and pepper. Whisk together.
Turn to low and slowly whisk in cheese. Mix until cheese is melted.
Place single layer of potatoes in casserole dish slightly overlapped. Spoon enough cheese mixture over the first layer of potatoes to cover them. Continue to layer the potatoes and cheese mixture alternating between the two.
Bake covered with aluminum foil over top for an hour. Bake uncovered for 20-30 minutes or until top is slightly browned. Cool for 5-10 minutes.