Which governor west of the Euphrates asked the Jews who had authorized them to rebuild the temple?

Answer: Tattenai

Tattenai and Shethar-bozenai challenged the builders, but God's eye was on the elders, so work went on while Darius was consulted.

“At that time Tattenai the governor of the region west of the Euphrates, Shethar-bozenai, and their associates went to the Jews and asked, “Who authorized you to rebuild this temple and restore this structure?””— Ezra 5:3 (BSB)