Heat oven to 400ºF. Butter a 9-by-13-inch baking dish and set aside.
Using a mandoline or the slicing attachment on a food processor, slice the potatoes approximately 1/8-inch thick. (If you don't want to slice all the potatoes at once, slice them one at a time and build the gratin as you go.) Create the first layer by laying the slices in overlapping rows in the baking dish. Once the first layer is down, season lightly with salt and pepper, then scatter with mushroom slices and a couple tablespoons of the cheese. (Don't over-do it on these layers, if you create a barrier between the adjoining potato layers, the gratin won't set.) Continue building layers until you're out of potatoes or out of room to build, but be sure to save 1/2 cup of the cheese for the top.
Pour 2/3 cup of the half-and-half over the gratin then spread both hands over the surface and push down to work the air out from the layers. Add remaining liquid only if half-and-half does not come to the surface when you push down. Sprinkle the gratin with cheese, cover loosely with foil, and place in middle of oven for 1 hour. Check for doneness by inserting the point of a paring knife straight into the gratin. If it goes through smoothly, remove the foil, return to oven, and turn on the broiler just long enough to turn the top golden brown. Remove and allow to sit at room temperature for 15 to 20 minutes before serving.