Pre-bake the casserole: Preheat the oven to 375 degrees F and butter a 9 x 13-inch casserole dish or other shallow 3-quart casserole dish.
Melt 1 tablespoon butter in a small saucepan over medium-low heat and add the shallot and garlic. Cook, stirring occasionally, until the shallot is softened but not browned, about 2-3 minutes. Remove from the heat and pour in the half and half.
Place a layer of potatoes in overlapping slices across the bottom of the casserole dish, then sprinkle lightly with salt and pepper. Sprinkle 1/4 of the cheese across the potatoes, then repeat 3 more times with the remaining potatoes, salt, pepper and cheese.
Pour the half and half gently over the potatoes.
Cover the dish with foil and bake for 30 minutes. Cool to room temperature, then refrigerate overnight.
Finish baking and serve: Preheat the oven to 425 degrees F. Uncover the potatoes and bake for 30 minutes, until the casserole is warmed through, bubbling, and lightly browned on top.