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Part 1.
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Put meat in a big heavy-bottomed pot . Add wine, 3 cups of port, the stock, peppercorns, fresh thyme and garlic. Stir. Bring to rapid boil over medium-high heat. Reduce heat and simmer uncovered, about 3 hours, stirring occasionally.
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After 3 hours, your pot should look kind of gross and sludge-y. The meat will be purple.
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Pull the meat out of the pot and set aside to cool. The broth should be thin and a deep mahogany color. Set it aside.
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When the meat is cool enough to handle, pull it off the bones. Discard the fat, bone, and gristle.
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Chop the meat into small pieces. Place in a bowl. Cover it and refrigerate.
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Part 2.
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Strain the braising liquid and discard the solids. Put the liquid in a large saucepan on the stove over high heat. Bring it to a boil, reduce the heat and simmer for 1 1/2 to 2 hours, or until it's reduced to about 3 cups.
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Put 3 tablespoons of butter and 3 tablespoons of flour in a small bowl.
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Mash them together into a paste.
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Whisk the butter/flour mixture into the reduced braising liquid over medium-high heat until it melts completely.
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Cook like this for 2-3 minutes, until the sauce has thickened.
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Remove the pan from the heat and set it aside.
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Part 3.
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Dice the carrots and shallots.
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Melt the remaining 5 tablespoons of butter in a large saucepan over medium heart.
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Toss in the carrots and shallots. Stir to coat in butter.
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Saute over medium heat for about 15 minutes, until the veggies are starting to get soft and translucent.
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Toss in the kosher salt, 1 1/2 teaspoons black pepper, and the white and brown sugars.
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Pour in the remaining 3 cups of port and all the red wine vinegar. Stir to combine.
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Cook over medium-high heat, stirring frequently, until the liquid has completely evaporated. This will take about 45 minutes.
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Part 4.
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Toss the meat into the pan with the veggies.
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Add the thickened braising liquid to the pan. Mix everything together.