Season the beef brisket with salt and pepper and coat it with flour, shaking off the excess. Heat the oil in a large, deep pan over medium heat. Brown the brisket lightly on all sides, for 5-8 minutes total time. Remove the meat and set it aside.
Discard all but 1 tablespoon of fat from the pan. Add the onions, carrots, and celery to the pan and cook 4-5 minutes. Add the garlic and cook 2 minutes more, stirring occasionally. Stir in the tomato paste and thyme and cook 3 minutes.
Uncover the pan and cook 30 minutes longer, or until the brisket is fork tender. Remove the pan from the oven and let the brisket rest in the pan juices for 15 minutes.
Transfer the brisket to a cutting board and slice it diagonally across the grain for optimal tenderness. Season the pan juices with salt and pepper as needed, and serve the brisket with some of the juices poured over the top and the rest passed at the table.