Check to see if the brisket fits into a Dutch oven or heavy-bottomed pot with a lid. If not, slice the brisket into 2 pieces against the grain to allow it to fit.
Put a cooling rack over a baking sheet. Put the brisket on the rack with the side lined with a thin layer of fat touching the rack. Sprinkle the salt and black pepper all over the brisket, making sure to season both sides. Let the brisket come to room temperature for about 30 minutes. (Tip: This step can be done up to 1 day in advance. Refrigerate the seasoned brisket, uncovered. By letting the brisket rest in the refrigerator, the sear will be more even and golden brown.)
Put a Dutch oven or heavy-bottomed pot over high heat. Add the oil and heat until shiny, about 30 seconds. Add the brisket fat-side up and sear until golden brown, 3 to 4 minutes. Turn the brisket over and sear the other side for another 3 to 4 minutes. Remove the brisket from the pot and set aside.
Reduce the heat to medium high and add the onions. Cook until just translucent, about 3 minutes. Pour the cola into the pot and scrape up any brown bits. Place the brisket back in the pot fat-side up, along with any juices that have collected around it. Bury the brisket under the onions. Add the beef stock, star anise, rosemary sprigs and bay leaf; the stock should not submerge the brisket, but come up about two-thirds of the way. Bring to a simmer and reduce the heat to medium low. Cover the pot and cook for 1 1/2 hours.
Remove the lid and flip the brisket over, making sure to nestle it back into the softened onions. Cover the pot and continue cooking until the beef is tender but not falling apart, an additional 1 1/2 hours. Remove the beef from the pot and set on a cutting board. Remove the rosemary stems, bay leaf and star anise and discard. Place the pot over high heat and cook the sauce until slightly reduced, about 10 minutes.
Thinly slice the brisket against the grain and spread onto a serving platter. Top the sliced brisket with the onions and spoon the sauce over the platter. Sprinkle with the parsley and thyme. Serve immediately.