Adjust oven rack to lower middle position and heat oven to 300°F. Line a 13 by 9 inch baking dish with two 24 inch long sheets of 18 inch wide (extra wide) heavy duty aluminum foil, positioning beets perpendicular to each other and allowing excess foil to extend beyond edges of pan.
Pat brisket dry with paper towels. Place brisket fat side up on cutting board, using dinner fork, poke holes in meat through fat layer about 1 inch apart. Season both sides of brisket with salt and pepper.
Heat 1 tsp oil in large skillet over medium high heat until oil just begins to smoke. Place brisket, fat side up, in skillet; weight brisket with heavy Dutch oven or cast-iron skillet and cook until well browned, about 7 minutes.
Remove Dutch oven, using tongs, flip the brisket and cook on second side without weight until well browned, about 7 more minutes. Transfer the brisket to a platter or cutting board.
Pour off all but 1 tablespoon fat from the pan (or if the brisket was lean, add enough oil to fat in skillet to equal 1 tablespoon); stir in onions, sugar, and ¼ tsp salt and cook over medium high heat, stirring occasionally, until onions are softened, 10 to 12 minutes.
Add garlic and cook, stirring frequently, until fragrant, about 1 minute; add tomato paste and cook, stirring to combine, until paste darkens, about 2 minutes. Add paprika and cayenne and cook, stirring constantly, until fragrant, about 1 minute. Add flour and cook, stirring constantly, until well combined, about 2 minutes. Add broth, wine, bay leaves, and thyme, stirring to scrape up browned bits from pan; bring to simmer and simmer for 5 minutes to fully thicken.
Pour sauce and onions into foil-lined baking dish. Nestle the brisket, fat side up, in the sauce and onions.
Fold foil extensions over and seal (do not tightly crimp foil because it must later be opened to test for doneness). Place in the oven and cook until fork slips easily in and out of the meat, 3 ½ to 4 hours (when testing for doneness, open foil with caution as the contents will be steaming). Carefully open foil and let the brisket cool at room temperature, 20 - 30 minutes.
Transfer the brisket to a large bowl; set a fine-mesh strainer over the bowl and strain the sauce over the brisket. Discard the bay leaves and thyme from the onions and transfer onions to a small bowl. Cover both bowls with plastic wrap, cut vents in plastic and refrigerate overnight.