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1
Combine all ingredients in a heavy-bottomed pot and bring to a boil. Cover and reduce the heat, then simmer for about 45 minutes. Remove the solids from the broth (use your fingers or a utensil, or strain the broth through a sieve) and discard. Use broth immediately, or cool and refrigerate (for up to 1 week) or freeze (for up to several months) for future use.
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2
Soak the dried mushrooms in the boiling water until they are soft. Lift the softened mushrooms from the liquid and gently squeeze them out, reserving the soaking liquid. Strain the liquid through a coffee filter to remove any grit, and set the soaking liquid aside. Roughly chop the softened mushrooms and set aside.
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3
Remove any soil or dirt from the fresh mushrooms with a soft brush or cloth. Trim the bottoms of the cremini or button mushrooms and slice them lengthwise. If using shiitakes, remove the stems and slice the caps. If using oyster, maitake, or chanterelles, gently tear them into small pieces.
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4
Combine the butter and olive oil in a very large wide skillet over medium heat until the butter is melted. Add the shallot and a pinch of salt and cook until softened and translucent.
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5
Add the softened dried mushrooms and stir well to coat them with the butter and oil. Cook for a few minutes, add the fresh mushrooms and another pinch of salt, and cook for 2 to 3 minutes, stirring to coat them with oil.
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6
Allow the fresh mushrooms to cook over medium heat, uncovered, until they are soft, shrinking, and beginning to brown.
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7
Clear a spot in the bottom of the skillet and add the tomato paste, allowing it to cook for a minute or two before stirring it through the mushrooms.
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8
Continue to cook until the mushrooms are browned at the edges, then add the vermouth, stirring through and scraping up any browned bits from the bottom of the pan.
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9
When most of the liquid has evaporated, add the reserved mushroom soaking liquid. Cook down until the mushrooms are glazed with the reduced liquid, stirring occasionally.
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10
Heat the broth in a large, heavy-bottomed pot and add the mushroom mixture to the hot broth. Puree with an immersion blender or in a blender or food processor until smooth.
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11
Add the sherry vinegar and thyme and stir through. Taste and adjust seasoning if necessary.
