Promoting Cross-border Energy Infrastructure Act This bill establishes a new process for approving the construction and operation of energy infrastructure across an international border of the United States and replaces the existing process established under specified executive orders. Specifically, this bill requires a person to obtain a certificate of crossing before constructing, connecting, operating, or maintaining a border-crossing facility for the import or export of oil, natural gas, or electricity across a U.S. border between Canada or Mexico. A certificate must be obtained from the Federal Energy Regulatory Commission (FERC) for a facility consisting of oil or natural gas pipelines or the Department of Energy (DOE) for an electric transmission facility. As a condition of issuing a certificate, DOE must require that an electric transmission facility be constructed, connected, operated, or maintained consistent with specified policies and standards. FERC and DOE must meet a deadline for issuing a certificate as set forth by this bill. The bill also requires FERC to meet a deadline for approving applications to import or export natural gas to or from Canada or Mexico.