I believe you are correct that special assessments for property improvements are added to the cost basis, and depreciated if this is rental property, and not expensed. Normally, you start to depreciate improvements when they are placed in service (when the work is done) regardless of when you paid for it. So I would list the work as a single $10,000 improvement performed either in 2022 or 2023, not as two split improvements.
As mentioned by others, this might qualify for bonus depreciation. I haven't educated myself on those rules so I can't comment directly.
