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1
Heat 1 tablespoon oil in a 10-inch well-seasoned cast-iron skillet over moderately high heat until hot but not smoking, then saute mushrooms and half of onion, stirring, until mushrooms are tender and browned, 8 to 10 minutes.
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2
Transfer to a bowl and wipe out skillet.
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3
Peel and coarsely grate apple and stir into mushrooms with thyme, 1/2 teaspoon salt, and 1/4 teaspoon pepper.
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4
Preheat oven to 450F.
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5
Peel potatoes and slice 1/8 inch thick (preferably using a mandoline or other manual slicer).
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6
Toss with remaining onion, 1/2 teaspoon salt, and 1/2 teaspoon pepper.
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7
Heat remaining 2 tablespoons oil in skillet over moderately high heat.
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8
Add one third of potato slices in 1 layer, overlapping slightly.
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9
Top with half of mushrooms, spreading in an even layer.
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10
Repeat layering with half of remaining potatoes, remaining mushrooms, and then remaining potatoes.
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11
Put skillet in middle of oven and bake 20 minutes.
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12
Invert a pizza pan or round platter over skillet and, holding pan or platter firmly against skillet, flip potato cake onto platter.
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13
Carefully slide cake back into skillet and press layers together with a spatula.
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14
Bake until underside is golden and crusty and potatoes are tender when pierced with a fork, 8 to 10 minutes more.
