is it possible for me to print out a list of the batches that I've recently submitted?
In the Progress section of your indexing home page, if you click Projects, you will see the last 5 you have worked on and how many records of each.
If you want to keep track in detail of the batches that you do, @J C Bingham has a beautiful spreadsheet setup that will help you do so. He shared it in/with the old Indexing Chat Group and might be willing to share it here. I have an early version of it, but I’ll let him decide whether and what to share.
Just to clarify, the information shown in the Projects tab of the Progress section shows the last five projects that you have worked on, not the last five batches that you have submitted. The Records Indexed number shown for each project is the sum of records indexed and records reviewed in that project that you have submitted.
I do have a set of spreadsheets that I use (thanks for the compliment @John Empoliti). In one I keep a record of the batches that I have submitted, in the other I keep a record of daily totals. If you are interested in them send me a private message and I will give you the Google Drive URL for them.