Same exact problem on same exact model. However, I did call Breville and they do actually sell parts for all their models. I took my entire unit apart and got the part number of the control PCBA for this unit.
Photos posted in the reddit post I made: https://www.reddit.com/r/ElectricalEngineering/comments/1abuzw9/clueless_me_looking_for_a_little_ee_help_pcba/?sort=top
When I called Breville, that PCBA cost is around $65 and will be in stock sometime around April 2024.
If anyone is trying to find solely the encoder (which Breville will not sell you), this is very very very close. Not sure it will work. I may just end up trying it...
https://www.tme.com/us/en-us/details/ec16e20-24p24c-sw/incremental-type-encoders/sr-passives/?utm_campaign=compare-2024-01-31&utm_medium=cpc&utm_source=octopart.com