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Preheat the oven to 350°. In a pie plate, bake the walnuts for about 8 minutes, or until lightly browned, then let cool and coarsely chop. Line a medium baking sheet with parchment paper.
Unfold the puff pastry and cut out four 4-inch rounds about 1/8 inch thick. Transfer the rounds to the prepared baking sheet and prick all over with a fork. Cover with another sheet of parchment paper and set a baking sheet on top. Bake the pastry for about 25 minutes, or until it is cooked and browned; remove the pastry from the oven. Increase the oven temperature to 400°.
Meanwhile, melt the butter in a large skillet. Add the onions and cook, stirring, over moderately high heat for 4 minutes; then cook over low heat, stirring often, until soft and golden, about 50 minutes; let cool. In a food processor, coarsely puree the onions and walnuts. Season with salt and pepper.
In a medium saucepan of boiling water, blanch the mushrooms until just tender, about 2 minutes. Drain, pat dry and slice 1/4 inch thick.
Spread the pastry rounds with the puree and arrange the mushrooms, overlapping, on top. Brush with olive oil and bake for 10 minutes, or until hot. Garnish with chives and serve.
The recipe can be prepared through Step 3 up to 1 day ahead. Wrap the pastry rounds in plastic and refrigerate the onion-walnut puree.
Frisée and herb salad.
A medium-bodied red with some tannin will contrast with the mushrooms. Consider one from Bordeaux or Tuscany.