Using fork, poke holes all over brisket. Rub entire surface of brisket with 1 tablespoon salt. Wrap brisket in plastic wrap and refrigerate for at least 6 or up to 24 hours.

Adjust oven rack to lower-middle position and heat oven to 325 degrees. Pat brisket dry with paper towels and season with pepper. Heat 2 teaspoons oil in 12-inch nonstick skillet over medium-high heat until just smoking. Place brisket fat side down in skillet; weigh down brisket with heavy Dutch oven or cast-iron skillet and cook until well browned, about 4 minutes. Remove Dutch oven, flip brisket, and replace Dutch oven on top of brisket; cook on second side until well browned, about 4 minutes longer. Transfer brisket to plate.

Heat remaining 2 teaspoons oil in now-empty skillet over medium heat until shimmering. Add onions and cook, stirring occasionally, until soft and golden brown, 10 to 12 minutes. Transfer onions to 13 by 9-inch baking dish and spread into even layer.

Combine cola, ketchup, onion powder, sugar, garlic powder, thyme, 1 teaspoon salt, and 1 teaspoon pepper in bowl. Place brisket fat side up on top of onions and pour cola ­mixture over brisket. Place parchment paper over ­brisket and cover dish tightly with aluminum foil. Bake until tender and fork easily slips in and out of meat, 3½ to 4 hours. Let brisket rest in liquid, uncovered, for 30 minutes.

Transfer brisket to carving board. Skim any fat from top of sauce with large spoon. Slice brisket against grain into ¼-inch-thick slices and return to baking dish. Serve brisket with sauce.