In a small bowl, combine the paprika, 1 tablespoon salt, and 1 tablespoon pepper. Rub the mixture all over the brisket. Let rest at room temperature for two hours or cover and refrigerate overnight (bring the meat to room temperature before cooking).

Position a rack in the center of the oven and heat the oven to 350 degrees F.

Meanwhile, heat the oil in a large pot over medium heat. Add the onions and garlic and cook, stirring occasionally, until very soft and pale gold, 15 to 20 minutes. Transfer the onions to a large heavy-duty roasting pan and spread them in an even layer. Set the brisket fat side up on the onions (it's okay if the pieces overlap), cover tightly with heavy-duty foil (or a double layer of regular foil), and braise in the oven for 1 hour. As the brisket cooks, it will give off quite a bit of liquid.

Pour the tomato puree and wine around the brisket and add the thyme sprigs. Cover and continue to braise the meat for 2-1/2 hours.

Add the carrots and mushrooms and continue to braise, covered, until the meat is fork-tender, about 1 hour more.

Transfer the meat to a cutting board and trim the fat. If using a whole brisket or a point half, separate the two layers of meat and trim the fat. With a slotted spoon, move the vegetables to a serving bowl.

Skim the excess fat from the pan juices, strain 2 cups of the juices, and bring to a boil in a small saucepan over medium-high heat. Boil until reduced to about 1 cup; the sauce should be rich and flavorful. Season to taste with salt and pepper.

Slice as much brisket across the grain as you need for the meal and serve with the vegetables and reduced sauce. Wrap the leftover brisket, vegetables, and juices separately. Leftovers will keep in the fridge for 3 to 4 days, or in the freezer for up to 2 months.

Serve with potato pancakes or mashed potatoes, and offer hot mustard, horseradish, and applesauce at the table.