Season brisket well with salt, pepper and thyme on both sides. Refrigerate to meld flavors for 2-24 hours.
When ready to cook, let the meat stand at room temperature for 30-60 minutes.
Heat 2-tablespoons oil in a large, heavy bottomed pot or Dutch oven over medium-high heat. Add brisket, fat side down. Cook without moving for 5 minutes until well browned. Then turn the brisket over and cook the other side until well browned, about 3-5 more minutes.
Remove and set the brisket aside on a plate. Then reduce the heat to medium and add the celery and onion and saute until golden, about 5 minutes, making sure to scrape up all the pieces stuck to the bottom. Add smashed garlic and saute another minute.
Then add the brisket back in. Pour wine over brisket and add 1/2 cup dried apricots. Bring to a boil, then cover and lower to a simmer over medium low heat.
Braise until brisket is very tender to the touch but not falling apart, about 3 1/2 to 4 1/2 hours. Cool brisket in the sauce and refrigerate overnight.
The next day, transfer the brisket to a large plate.
Strain braising liquid into a large bowl to remove large chunks (you may have to heat up the liquid first if a hard fat layer has formed.
Return the liquid to pot, bring to a simmer, add remaining apricots and cook until apricots are soft and sauce is thick, about 20 minutes.
Cool slightly and adjust with salt and pepper to taste.
Slice brisket against the grain and serve with sauce and apricots!
Note: Cook brisket a day before you intend to serve.