His father was a mathematics professor at the University of Florence, while his mother was an Englishwoman who had worked as a tutor to members of the Corsini family.

He emigrated to Australia at the age of 21, initially spending time at the Palmer River Goldfields in the colony of Queensland.

[1] After two years working in Italy and London, Fiaschi returned to Australia in 1879 and established a medical practice in Windsor, New South Wales.

He moved back to Sydney in 1883 and became active in the newly created New South Wales branch of the British Medical Association (BMA), serving a term as president from 1889 to 1890.

[1] Fiaschi applied the antiseptic surgery principles of Joseph Lister and translated Italian surgeon Edoardo Bassini's works on hernia repair into English.

[1] In 1876, Fiaschi married Catherine Ann Reynolds, a nurse and former nun with whom he had worked at St Vincent's Hospital.

[1] In 1968, Fiaschi's daughter Clarissa donated a replica of Florence's Porcellino statue to the City of Sydney in honour of her father.