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1
Position a rack in the center of the oven and preheat it to 300u00b0F.
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2
If needed, trim the excess fat from the brisket so that there is about a 1/4-inch-thick layer remaining. Pat the brisket dry with paper towels and season it generously with salt and pepper. Select a Dutch oven or stainless-steel roasting pan that will be large enough to accommodate the brisket, vegetables, and braising liquid. Heat the pan over medium-high heat, and then add the oil. When the oil just begins to smoke, add the brisket, fat side down first, and brown it on both sides, 4 to 6 minutes per side. Remove the brisket from the pan and set aside.
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3
Carefully pour off and discard all of the fat from the pan. Quickly add the cider, wine, thyme sprigs, garlic, bay leaves, and 1 teaspoon salt and scrape the bottom of the pan to release any browned bits. Return the meat to the pan, fat side up, with any accumulated juices. Nestle the meat into the liquid so it is nearly covered. Increase the heat to high and bring the liquid to a boil. Cover the pan tightly with a lid or aluminum foil and transfer it to the oven.
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4
Cook the meat for 1 1/2 hours, and turn the meat over in the pan. Cover and return the brisket to the oven. After another 1 3/4 hours of cooking, add the squash and onions to the pan, nestling them under and around the brisket. Continue cooking until the squash and onions are tender when pierced with a fork and the meat is very tender and easily shreds, 30 to 45 minutes longer. To test the brisket for doneness, use two forks to gently pull the meat apart in the center. If it is still a bit tough but the squash and onions are done, transfer the vegetables to an ovenproof dish using a slotted spoon; set them aside, covered with aluminum foil to keep warm. Continue to braise the brisket for about 15 minutes more, and then retest. The internal temperature should be 195u00b0F for lean brisket, or 205u00b0F if it is a fattier brisket.
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5
When the brisket is done, transfer it to a cutting board, fat side up, while you finish the sauce. Decrease the oven temperature to 200u00b0F. Remove the butternut squash and onions from the braising liquid using a slotted spoon and place them in an ovenproof dish (if you did not do this earlier). Taste the vegetables and season them with salt, if needed. Cover the dish and put the vegetables in the oven to stay hot.
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6
Strain the braising liquid into a small bowl, discarding the solids. Allow the fat to separate, and then skim and discard it. Clean out the pan and pour the braising liquid back in. Bring the liquid to a boil over high heat, and cook, stirring occasionally, until it is reduced to a thickened sauce that coats the back of a spoon, 15 to 20 minutes. Taste and adjust the seasoning with salt and pepper.
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7
Cut the brisket against the grain into 1/2-inch-thick slices, or, if it is too tender to slice, pull it apart into large chunks. Arrange the vegetables on a large serving platter, with the sliced brisket in the center. Spoon the sauce over the brisket and vegetables. Garnish with the minced thyme. Serve immediately, passing any remaining sauce at the table.
