Cut the tails and leaves from the beets but leave about 1 inch of the stems attached. Wash well and put them in a 2-quart saucepan with enough cold water to cover. Bring to a boil over high heat, reduce the heat to medium low, and simmer until tender when pierced with a skewer, 30 to 35 minutes. Drain and rinse with cold water to cool. When cool enough to handle, slip off the peels. Slice the beets 1/4 inch thick.

While the beets cook, put the potatoes in a 4-quart saucepan with enough water to cover and 1 tablespoon salt. Bring to a boil over high heat, reduce the heat to medium low, and simmer until the potatoes are tender when pierced with a skewer, 20 to 25 minutes. Drain and rinse with cold water to cool. When cool enough to handle, peel and slice 1/4 inch thick.

In a small bowl, whisk the sour cream, horseradish, and vinegar. Mix in the apple and onion, if using. Season to taste with salt, pepper, and vinegar.

In a medium bowl, toss the fennel and parsley with 2 tablespoons of the horseradish dressing, the olive oil, 1/4 teaspoon salt, and a pinch of pepper.

Arrange the brisket, potatoes, and beets on 4 large plates. Sprinkle lightly with salt and drizzle with some of the dressing. Pile the fennel on top and serve the remaining dressing on the side.

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.