Yes, an interesting period in French history (one of many!). Henri III had his hands full, not least with the Wars of Religion.
Nevertheless, he found time to reform the French monetary system. Part of the reform was to introduce the franc
as a denomination (or rather re-introduce, as it had been used in the 14th century). The main idea was to abandon the livre
as money of account and introduce the écu
in its place. The value of one écu was fixed to 60 sous tournois and one franc to 20 sous tournois. There were coins with denominations of 1
and 1/8 écu
as well as 1
and 1/4 franc
. In addition, there were various smaller denominations of sous
. Overly complex?
Yes... The system lasted until 1602, when Henri IV decided it was time to go back to the livre and flexible denominations (that is, that could be changed at the king's discretion).
Looking again at your coin, it might be a one franc that has been clipped rather than a half franc. If it is important to know which, an accurate measurement of diameter and weight should make it possible to determine with certainty.