We are a U.S. 14 CFR Part 135 operator. Do I need to hold an authorisation if I intend flying commercially to Europe?

Yes. Part-TCO is fully applicable to flights conducted under an U.S. 14 CFR Part 135 certificate. Note: The international standards laid down in the ICAO Annexes exceed the requirements of U.S. 14 CFR Part 135 in several aspects, including but not limited to crew training requirements and the implementation of Safety Management Systems.