Trim the brisket's fat cap to about 1/4 inch. Cut a small corner off straight across the grain to mark the slicing direction. Pat dry and season all over with the salt and pepper.
Heat the oil in the slow cooker on Sear/Sauté (or in a large skillet over medium-high heat). Brown the brisket on both sides, 4 to 5 minutes per side. Move it to a plate.
Add the onions and a pinch of salt to the drippings. Cook over medium heat about 10 minutes, stirring and scraping up the browned bits, until soft and starting to turn golden.
Stir in the garlic and tomato paste and cook 1 to 2 minutes. Add the wine, scrape the bottom, and simmer 1 minute. Stir in the broth, the tomato sauce, vinegar, brown sugar, and Worcestershire until the sugar dissolves.
Scoop the onions and sauce into a bowl. Spread the carrots in the bottom of the slow cooker. Set the brisket on top, best-browned side up. Pour the onions and sauce over it and tuck in the thyme and bay leaves. Don't add more liquid.
Cover and cook on HIGH about 5 hours, or LOW 8 to 10 hours, until a fork slides into the thickest part with very little resistance.
Move the brisket to a cutting board, tent with foil, and rest 20 to 30 minutes. Discard the bay leaves and thyme stems. Lift the carrots and onions into a bowl with a slotted spoon and keep warm.
Skim the fat from the cooking liquid. Pour the liquid into a saucepan and bring to a simmer. Whisk the potato starch into the cold water, whisk it into the simmering liquid, and remove from the heat as soon as it thickens. Taste and add salt if needed.
Slice the brisket parallel to the cut corner (across the grain), about 3/8 inch thick. Arrange on a platter with the carrots and onions and spoon gravy over the top. Serve the rest of the gravy on the side.
Notes
Regular slow cooker: Sear the brisket and cook the onions and sauce in a large skillet, then layer everything in the slow cooker.
Flour instead of potato starch: Skip the slurry. Sprinkle 2 tablespoons all-purpose flour over the onions after the tomato paste and stir 1 minute before adding the wine. The cooking liquid will thicken as it cooks.
Ketchup instead of tomato sauce: Use ketchup in place of the tomato sauce and vinegar, and cut the brown sugar back a little, since ketchup is already sweet.
Milder garlic: Use about a third of the garlic.
No wine: Replace it with more beef broth.
Onion soup mix: Stir a packet into the sauce and leave out the salt.
Make ahead: Cook, cool about 1 hour, and refrigerate the whole brisket in its liquid overnight. Lift off the fat, slice cold, make the gravy, pour it over the slices in a baking dish, and reheat covered at 325°F for 30 to 40 minutes.
Keep warm: Lay the slices back over the onions and carrots in the slow cooker, pour the gravy over, cover, and hold on Keep Warm for up to 2 hours.
Storage: Refrigerate leftovers in the gravy, covered, up to 4 days, or freeze up to 3 months. Thaw overnight in the fridge.
Reheating: Reheat covered in the gravy in the microwave or a 325°F oven until hot.