When you go into the Change Class menu, you will now see a list of cards drawn from each class. This list is compatible with existing saved games and requires no additional saved data. It is created by taking the starting deck and subtracting the cards the character still has.
There is a known bug with this implementation. If you have a saved game where characters were created using old versions of a level-up deck, the list of drawn cards will include cards that were later added to the starter deck, even though the character never drew those cards. We feel this is an acceptable trade-off for adding this feature.
Thank you to Sarios8781 on the Discord for pointing out the simplest route to implementation of this oft-requested feature.