Heat your oven to 350°F (175°C).

Trim your brisket, if necessary, so that only a thin layer of fat remains — about ⅛- to ¼-inch thick. Season it generously on top with salt and black pepper.

Heat a large Dutch oven over high heat with 1 to 2 tablespoons of oil until very hot. Place the whole brisket seasoned side-down into the hot pan then season the second side with more salt and pepper. Cook until crusty and well-browned on both sides, about 5 to 7 minutes per side.

Halve and peel each onion, and slice a generous ¼-inch thick.

Add the onions to your empty pot, season with salt and pepper, and cook, stirring frequently, until the onions have softened and developed a rich brown color, 10 to 15 minutes. Add the garlic and cook for another minute.

Remove the pot from the heat. Place the browned brisket on the onions and pour any juices that accumulated on the plate over it, plus 1 ½ cups of water. Spread the tomato paste over the top of the brisket. Sprinkle with additional black pepper and any leftover salt. Scatter the carrots in the pan. Cover the pot and transfer to the oven.

Braise your brisket: Let the brisket cook, untouched, for 1 ½ hours then remove from the oven. Carefully transfer the brisket to a cutting board and thinly slice across the grain. Return the slices to the pot, leaning them slightly so that you can see a bit of the top edge of each slice. If the liquid looks low, add another ½ cup of water. Replace the lid and return it to the oven for an additional 1 ½ to 2 ½ hours.

If serving the next day, let the brisket cook for 3 hours then remove from the oven. If the liquid looks low, add another ½ cup of water. The brisket is done when you can easily pull a slice apart with a fork.

Chill brisket overnight, or for two nights. About 1 hour before serving, heat oven to 350°F (175°C). Remove the lid and use a spoon to remove any solidified fat. Carefully transfer the brisket to a cutting board and thinly slice across the grain. Return the slices to the pot and warm in the oven for 35 to 45 minutes.

Serve the sliced brisket, spooning onions and sauce from the pan over each plate.