For breast cancer patients who receive chemotherapy first and then get surgery, long-term survival is longer for women who do not have any microscopic cancer at the time of surgery. The primary objective of this study is to assess whether a combination of chemotherapy and erlotinib will result in no evidence of microscopic disease (pCR) at the time of surgery in greater than 20% of enrolled subjects. After completing all chemotherapy, patients will also receive maintenance erlotinib for 12 months. This is given to study the tolerability of maintenance Erlotinib and also to evaluate if maintenance erlotinib will decrease the rate of tumor recurrence.