Using fork, poke holes all over brisket. Rub entire surface of brisket with 1 T 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. Pat brisket dry with paper towels and season with pepper. Heat 2 t oil in 12 inch nonstick skillet over med-high til just smoking. Place brisket fat side down in skillet; weigh down with dutch oven or heavy cast iron skillet and cook til well browned, about 4 minutes. Remove dutch oven, flip brisket and replace dutch oven on top and cook til other side is well browned,about 4 minute Transfer to platter.
Heat remaining 2 t oil in same skillet over med til shimmering. Add onions and cook, stirring occasionally, until soft and golden brown, 10-12 minute Transfer to 13x9 inch baking dish and spread into even layer.
Combine cola, ketchup, onion powder, sugar, garlic powder, thyme, 1 t salt, and 1 t pepper in a 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 foil. Bake til tender and fork slips easily in and out of meat, 3 1/2 - 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 1/4 inch thick slices and return to baking dish. Serve with sauce.
To make ahead: Follow recipe through step 4. Allow brisket to cool in sauce, cover, and refrigerate overnight or up to 24 hours. To serve, slice brisket, return to sauce, and cover with parchment paper. Cover baking dish with foil and cook in 350 degree oven til heated, about 1 hour.