How to Make It
Preheat the oven to 350°. On a medium baking sheet, toss the sliced mushrooms with 2 teaspoons of the olive oil and season with salt and pepper; spread out in a single layer. Bake for about 17 minutes, turning occasionally with a spatula, until browned and crisp.
Slice the potatoes 1/8 inch thick with a mandoline or a thin knife. Rinse the slices in a bowl of cold water, changing the water twice, to remove the potato starch; drain. Spread the slices in a single layer between paper towels and pat dry.
In a large skillet, heat the vegetable oil until almost smoking. Add half of the potato slices and cook over moderately high heat, turning once, until golden and crisp, about 7 minutes. Remove the chips with a slotted spoon and transfer to paper towels to drain. Repeat with the remaining potato slices.
In a large bowl, whisk the remaining 1/4 cup of olive oil with the sherry vinegar and 1/2 teaspoon of salt; season with pepper. Add the frisée, tomatoes, scallion, shallot, shiitakes and potatoes and toss well. Let stand for 15 minutes for the flavors to combine. Add the crab, divide among 4 plates and serve.