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1
Clean and trim mushrooms; if using shiitake, remove stems. Rough chop mushrooms into pieces a little smaller than bite sized.
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2
Heat saucepan over medium heat, add olive oil. When pan is hot, add sliced onion, spreading out in a single layer as much a possible. Allow onions to cook without stirring until fragrant, about 7 minutes. Lower heat one notch, adding a pinch of salt and a pinch of sugar to encourage caramelization. Continue to cook, stirring occasionally, for at least 20 minutes (depending on your onions and your preference).
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3
When onions are golden brown and edges are nicely caramelized, add mushrooms along with thyme leaves. Continue to cook, stirring occasionally, for about 6 minutes. At this point, having released their juices, mushrooms should be reduced in size and golden brown. Add salt to taste, stir in minced parsley and transfer mixture to a bowl. Set aside and allow mixture to cool.
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4
Add red wine, crushed garlic cloves and thyme sprigs to heated saucepan (same one you caramelized onion/mushrooms in). Bring to a boil and reduce to a simmer. Continue to simmer, stirring occasionally, until volume has been reduced by half, about 20 minutes. (Cover cooled onion/mushroom mixture with plastic wrap and place in refrigerator).
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5
Add chicken broth, Worcestershire sauce and beef broth to reduced wine. Add several grinds of fresh black pepper. Continue to simmer, stirring occasionally, until volume has been reduced by half, about 20 minutes. Strain reduction through a sieve into a small bowl or jar and let cool. Discard herbs and garlic. Once cool, cover with plastic wrap (or lid if using a jar) and place in refrigerator.
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6
When ready to grill burgers, prepare grill (can also use a grill pan). Remove onion/mushroom mixture from refrigerator and set aside to come up to room temperature. Pour reduction into saucepan and slowly heat up on low. Add one tablespoon of unsalted butter to reduction and stir to combine.
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7
In a shallow baking pan, combine meat, 1/2 teaspoon of salt and black pepper to taste. Divide into four equally sized portions. Add one tablespoon of crumbled blue cheese to each portion of meat. Without packing tightly, carefully form four patties, distributing crumbled cheese throughout each burger. Make a slight indentation in the center of each burger.
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8
Once grill is hot, cook burgers to desired doneness - if your grill is really hot, about 3 minutes per side for rare, a minute longer for medium. Note that this burger tastes best when burgers are medium-rare and the center is still a little pink.
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9
Spoon desired amount of onion/mushroom mixture onto burgers and top with 1 1/2 tablespoons (to start at least) of red wine reduction. Enjoy immediately.
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10
Leftover reduction can be refrigerated and used within a day or two.
