Place chicken breasts between two sheets of wax paper on a solid level surface. Firmly pound chicken with the smooth side of a meat mallet to a thickness of ½-inch. (If you don't have a meat mallet, you can slice the chicken to make thin fillets).
Season chicken thoroughly with salt & pepper.
In a large nonstick skillet, heat the olive oil to medium-high heat.
Add the chicken and reduce heat to medium.
Cook, turning once until nicely browned and opaque throughout, about 4-5 minutes per side.
Transfer chicken to a plate and cover with foil to keep warm.
If there is no oil left on the pan, add one tablespoon of olive oil.
Add the shallots and garlic and cook, stirring often until softened. a bout 2 to 3 minutes.
Add the tomatoes and vinegar. Saute stirring often for 2 minutes. Return the chicken to the pan and cook until the tomatoes begin to soften and split. About 2 more minutes.
Add the basil and salt & pepper to taste.
Serve warm with crusty bread for dipping in the balsamic reduction.