Hi @tony_le3
You can achieve this functionality by using any Order extended Grid module. Such a module will give you the ability to add extra columns in your order grid backend and you can customize one of the columns to be a 'Printed' label.
Alternatively, you can use an Order Grid Labels module. Such a module will also help you add ~n number of extra columns in your order grid. You can create a new label with this module and set it as "Last Printed" and add a small printer icon inside that label. You can use this to add the print logo on the orders which you have last printed.
I hope this helps! Please mark it as an acceptable solution.
Cheers!
Jojo