Use DateList and extract the year-month-date:
(DateList /@ {"3/8/2020", "3/9/2020"})[[All, {1, 2, 3}]]
(* {{2020,3,8},{2020,3,9}} *)
EDIT: Sorry -- I didn't notice the inner brackets. Still supposing the date is a string:
(DateList[#[[1]]] & /@ {{"3/8/2020"}, {"3/9/2020"}})[[All, {1, 2, 3}]]
(* {{2020,3,8},{2020,3,9}} *)
EDIT 2:
Switch off Warning::ambig to suppress warning messages, then switch back on if you like:
Off[DateList::ambig]
newList = (DateList[#[[1]]] & /@ {{"3/8/2020"}, {"3/9/2020"}})[[All, \
{1, 2, 3}]]
(* {{2020,3,8},{2020,3,9}} *)
On[DateList::ambig]