Heat the oil in a pan over medium high heat. Add in shitake mushrooms and shallots. Season with salt and pepper and cook until the mushrooms are tender and start to turn golden.
Preheat the oven to 350°F and grease a 9- by 12-inch gratin dish with oil. Slice the potatoes as thinly as you can and arrange them in a layer, overlapping the edges slightly like shingles. Sprinkle the potatoes with salt and freshly ground pepper, a third of the cheese and a third of your mushrooms. Repeat 2 to 3 more times.
3. Carefully pour the cream over the potatoes. Dot the top of the gratin with the three tablespoons of butter and bake it for about an hour. Halfway through the baking time, take the gratin dish out of the oven and gently press the potatoes flat with a spatula to keep the top moist. Sprinkle the remaining cheese on top of the gratin for the last 15 minutes of baking. The gratin is done when the potatoes are soft and the top is golden brown.