Step 1

Place porcini and 1 cup boiling water in medium bowl. Place small bowl atop mushrooms to keep submerged. Let soak 20 minutes. Drain and coarsely chop mushrooms.

Step 2

Melt butter with oil in medium skillet over medium heat. Add mushrooms and sauté until beginning to brown, about 3 minutes. Sprinkle with salt and pepper. Remove from heat. Whisk 1/4 cup Parmesan and next 4 ingredients in small bowl; season with salt and pepper. DO AHEAD: Mushrooms and cheese mixture can be prepared 1 day ahead. Cover separately and chill.

Step 3

Preheat oven to 325°F. Butter wide shallow 2-quart baking dish. Arrange 1/4 of potato slices in bottom of dish. Sprinkle lightly with salt and pepper. Scatter 1/4 of mushrooms over. Repeat. Spread half of cheese mixture over, shaking dish to settle. Repeat with remaining potatoes and mushrooms in 2 layers each; spread remaining cheese mixture over. Sprinkle 2 tablespoons Parmesan over. Place gratin dish on rimmed baking sheet.

Step 4

Bake gratin until top is brown and sauce is bubbling at edges, about 1 hour 15 minutes. Let gratin rest 15 minutes before serving. DO AHEAD: Can be made 2 hours ahead. Let stand at room temperature. Tent loosely with foil and rewarm in 300°F oven 20 minutes.

Step 5

*Available in the produce section of many supermarkets and at specialty foods stores and Italian markets.

Step 6

**Italian cream cheese; sold at many supermarkets and at Italian markets.