Preheat your oven to 325°F (165°C). Season your brisket generously with salt and pepper on both sides. Don't be shy—this is a big piece of meat.
Heat olive oil in your Dutch oven over medium-high heat until it's shimmering but not smoking.
Sear the brisket on both sides until deeply browned, about 4-5 minutes per side. This isn't the time for pale meat. We want caramelization, people!
Remove the brisket to a plate and lower the heat to medium. Add onions and carrots to the pot, cooking until they start to soften, about 5 minutes.
Add the garlic and cook for another minute until fragrant. Your kitchen should smell amazing right about now.
Pour in the red wine, scraping the bottom of the pot with a wooden spoon to release all those flavorful browned bits. Let it simmer for 2-3 minutes.
Add the beef broth, tomato paste, bay leaves, thyme, and brown sugar. Stir to combine and bring to a gentle simmer.
Return the brisket to the pot, fat side up, along with any juices that collected on the plate. The liquid should come about halfway up the sides of the meat.
Cover the pot with a tight-fitting lid and transfer to the preheated oven.
Cook for 3-4 hours, until the meat is fork-tender. Seriously, you should be able to pull it apart with minimal effort.
Remove the brisket to a cutting board, tent loosely with foil, and let it rest for 20 minutes.
Meanwhile, strain the cooking liquid into a saucepan if you want a smooth sauce (optional). Simmer over medium heat until reduced by about one-third.
Whisk in the butter for richness and glossiness. If you want a thicker sauce, add a slurry of 2 tablespoons flour mixed with 2 tablespoons cold water.
Slice the brisket against the grain into thin slices. Seriously, against the grain—it makes all the difference.
Serve with the red wine sauce poured over the top. Watch as people close their eyes in satisfaction with the first bite.