Brogi was born in Cairo, Egypt. He was became a priest on 5 May 1963. Pope John Paul II made Brogi titular archbishop of Città Ducale, Apostolic Nuncio to Sudan, and Apostolic Delegate to Somalia on 13 December 1997. Pope John Paul named him Apostolic Nuncio to Egypt and Apostolic Delegate to the Arab League on 5 February 2002.