U.S. #554
Series of 1922-25 2¢ Washington
Flat Plate Printing
Issue Date: January 15, 1923
First City: Washington, DC
Quantity Issued: 21,681,541,977
Printed by: Bureau of Engraving and Printing
Printing Method: Flat plate
Perforation: 11 gauge
Color: Carmine
By the way, in case anyone wants to look up these statistics for other stamps, I found that information in the stamp listing on the Mystic Stamps web site. Just put the Scott catalogue number of your stamp of interest in the search box: https://www.mysticstamp.com/Products/United-States/213/USA/
It depends on how you define an issue. You say you have a huge pile of flags, but there are likely several Scott catalog numbers. For the 2007 flag issue, there were seven different non-denominated and six different denominated major catalog numbers. Back in the 3rd Bureau there were LOTS of different catalog numbers for the various 2c stamps. According to Max Johl's book, considered as definitive by many, including me, there were 73,163,469,678 2c Washington stamps issued during the 3rd Bureau, but that includes flat plate, offset, rotary presses, and includes booklets and coils. Sheet stamps only were over 59 billion, but Johl doesn't split out TWO CENTS from 2 CENTS.
Scott provides an answer to the most common Commemorative on pages 571-576 of the 2016 U.S. Specialized: