Preheat oven to 325F.
"I have a pretty well-equipped kitchen, with lots of really big pots and Dutch ovens at home," says Rachael. "But, I don't have one big enough to cook a huge brisket like this one. If you do, leave the brisket whole. If not, after you trim the brisket of some of the fat, cut it in half widthwise. "
Place your largest Dutch oven over medium-high heat with 3 turns of the pan of EVOO, about 3 tablespoons. While the oil is heating up, liberally season the brisket first with the pepper and then with the salt -- don't be shy about seasoning the meat. Once the oil has started to ripple, add whole brisket or one of the halves to the pot and brown thoroughly on one side before turning over onto the second side. If brisket was halved, once the first piece has browned, turn it on its small side, standing it up against the side of the pot so that you are browning every little inch. Once the first half is standing up on its end, add the second half of the brisket to the pot and repeat the same process, working both pieces at the same time.
Once both pieces are nice and brown, remove them from the pot and reserve. Add sliced onions, chopped carrots, chopped celery, crushed garlic and thyme. Cook for about a minute, stirring to scrape up all of the bits on the bottom of the pot. Add tomato paste and cook for another minute or so. Add wine, chicken stock, parsley and browned brisket back to the pot.
Nestle the brisket down into the liquids and bring up to a simmer. Place a lid on the pot and transfer to the oven. Don't forget to bend your knees -- this pot is heavy and no meal is worth throwing your back out for!
Braise for 2 hours then add the whole carrots to the pot. Return the lid and continue to braise for another hour. Remove the pot from the oven and stick the brisket with a fork at its thickest part. If the fork easily slips out, the brisket is tender and ready. If not, return it to the oven for another 20 minutes and check it again. Once it is tender and out of the oven, let it rest in the pot with all the liquids for about 15 minutes. Remove brisket from the pot to a cutting board and slice it into about 1/4-inch thick slices with a long, thin, sharp knife against the grain of the meat. Serve a few slices of brisket topped with the sauce and onions with the carrots alongside. Serve with crusty bread, smashed potatoes, orzo or even spaetzle -- anything that helps sops up the sauce.