Preheat your oven to 325 degrees
Pat the brisket dry and sprinkle liberally on both sides with salt and pepper. Heat a heavy roasting pan or enameled cast-iron pot over medium high heat, using two burners if your pot is on the larger side. Add the oil and heat until hot. Sear the mean, fat side down, until golden brown, and crusty brown, about 10 minutes. Carefully flip the brisket and sear on the other side for another 5-7 minutes. Remove the brisket to a large plate or platter or baking sheet. Remove all but 2 tablespoons of fat from the pan.
Turn the heat down to medium-high heat add the sweet onions to the pan and cook, scraping the bottom until just starting to brown, 4 to 5 minutes. Add the wine and cook until reduced by about a third, about 4 minutes. Add the chicken broth, garlic cloves and thyme sprigs. Return the brisket and any juices to the pan, cover tightly with foil, transfer to the oven and cook for 1 1/2 hours.
Carefully transfer the brisket to a cutting board and slice the meat across the grain into approximately 1/8-1/4 inch thick slices. Remove the thyme sprigs. Return the brisket slices back to the pot, overlapping them at an angle so that you can see a bit of the top edge of each slice. The end result should resemble the original unsliced brisket leaning slightly backwards. Cover the pot tightly and return to the oven.
Lower the heat to 325 degrees and cook the brisket until fork tender, 1 3/4 to 2 1/2 hours, or longer if necessary. At this point, you can serve the brisket or refrigerate overnight (see below for storage and reheating instructions).