In a large heavy skillet heat the oil over moderately high heat until it is hot but not smoking and in it cook the chicken breasts, patted dry and seasoned with salt and pepper, skin sides down, for 3 to 4 minutes, or until they are golden. Turn the breasts and cook them over moderate heat, covered, for 7 to 8 minutes more, or until they are cooked through. Transfer the breasts to a platter and keep them warm. To the skillet add the garlic, the shallot, and the 2 thyme sprigs (or the dried thyme if using) and cook the mixture, stirring, until the shallot is softened. Add the vinegar, boil the mixture until the vinegar is almost evaporated, and add the mushrooms. Cook the mixture over moderate heat, stirring, for 1 minute, add the wine, and boil the mixture until the wine is almost evaporated. Add 1 1/4 cups of the broth and the scallions and simmer the mixture for 1 minute. In a small bowl whisk together the remaining 1/4 cups broth and the arrowroot and add the mixture to the skillet with the tomatoes and he fresh thyme leaves. Simmer the sauce, stirring, for 1 minute, or until it is thickened, stir in any juices that have accumulated around the chicken, and add salt and pepper to taste. Transfer a chicken breast to each of 6 dinner plates, spoon some of the sauce over each breast, and garnish each serving with 1 of the additional thyme sprigs.