Roughly, pick a date and start writing all new data to both, while running etls to backfill data from old to new. Once that is done, you use a feature flag to do a small % of reads from the new system and wait and pray. If nothing major pops up, you slowly ramp up the % of reads until you're confident (as much as one can be) that everything is working, then you move 100% of reads to the new system. Finally, you turn off writes to the old system, clean up the feature flag and remove any old unneeded code.

jk of course you never do any of the other stuff after turn off writes to the old system you just leave the FF turned on to 100% and never think about it again :-p