Pat the roast dry with paper towels and season all over with 2 teaspoons of kosher salt and 1/2 teaspoon of black pepper.

Heat 1 tablespoon of olive oil in a large Dutch oven over medium-high heat until shimmering. Add the roast and sear until dark golden-brown on two sides, about 4 to 5 minutes per side. Transfer to a large plate.

Reduce the heat to medium. Add the remaining 2 tablespoons olive oil, sliced onions, the remaining 1 teaspoon kosher salt, and remaining 1/2 teaspoon black pepper to the pot. Stir to combine. Cover and cook undisturbed for 5 minutes. Uncover, stir, cover again, and cook for another 5 minutes.

Uncover and continue cooking, stirring occasionally at first and then more frequently once the onions start to brown, until they are a rich caramel brown color, about 30 to 40 minutes. Reduce the heat as needed if the bottom of the pot gets too dark, scraping the bottom when stirring.

Stir in the minced garlic and cook for 1 minute. Sprinkle the flour over the onions, stir well, and cook for 1 minute. Stir in the wine and scrape up the browned bits from the bottom of the pot. Stir in the beef broth, thyme sprigs, and bay leaves.

Return the roast and any accumulated juices to the pot and bring to a simmer. Cover and transfer to the oven. Cook until the meat is gently falling apart when pricked with a fork, about 3 to 3 1/2 hours.

Skim off some of the fat from the surface if desired. Remove and discard the thyme stems. Shred the roast into bite-size pieces in the pot with tongs before serving. Garnish with more fresh thyme leaves if desired.