Season the chicken breasts on both sides with salt and pepper. Dredge in the flour and shake off any excess.
Combine the oil and 1 T. of the butter in a large saute pan over medium-high heat. When the oil is hot, place the chicken in the pan and cook, turning once, for 4 to 5 minutes on each side, until golden brown and cooked through. Remove from the pan and keep warm.
Add the shallot and garlic to the hot pan; saute for 1 minute, then add the vinegar to deglaze the pan, using a wooden spoon to scrape up all the browned bits on the bottom. Add the chicken broth and tomatoes and cook for 1 to 2 minutes, until reduced by half. Remove the pan from the heat. Stirring constantly, add the basil, remaining 2 T. butter and salt and pepper to taste. Stir until the butter is incorporated into the sauce.
Place a chicken breast on each plate and pour the sauce over. Serve at once.