This research study is a Phase II clinical trial. Phase II clinical trials test the effectiveness of an investigational drug to learn whether the drug works in treating a specific cancer. "Investigational" means that the drug is still being studied and that research doctors are trying to find out more about it-such as the safest dose to use, the side effects it may cause, and if the drug is effective for treating different types of cancer. It also means that the FDA has not approved this drug for use patients undergoing adjuvant treatment for HER2+ breast cancer. Trastuzumab emtansine (T-DM1) is a drug that may stop cancer cells from growing. This drug has been used in other research studies and information from those other research studies suggests that this drug may help to prevent the recurrence of breast cancer in this research study. The use of T-DM1 in this research study is experimental, which means it is not approved by any regulatory authority for the adjuvant treatment of HER2-positive breast cancer. However, it FDA-approved for metastatic HER2-positive breast cancer. T-DM1 has caused cancer cells to die in laboratory studies. In preclinical studies, this drug has prevented or slowed the growth of breast cancer. The breast cancer treatments (paclitaxel and Trastuzumab) used in this study are considered part of standard-of-care regimens in early breast cancer. A standard treatment means that this is a treatment that would be accepted by the majority of the medical community as a suitable treatment for your type of breast cancer. In this research study, the investigators are looking to see if the study drug T-DM1 will have less side effects than traditional HER2-positive breast cancer treatment of trastuzumab and paclitaxel. The investigators are also hoping to learn about the long term benefits and disease-free survival of participants who take the study drug T-DM1 in comparison to those participants to take the combination of trastuzumab and paclitaxel.