Availability Group database stuck in Not Synchronizing / Recovery Pending mode
Since the server had been offline for a while we thought it may have gone outside the recovery window of the primary. We decided to try applying the latest transaction logs on the database to see if that would kick-start the recovery process:

-- Remove database from Availability Group:    
Alter Database [StackExchange.Bicycles.Meta] SET HADR OFF;

-- Apply t-logs to catch up. This can be done manually in SSMS or via:
RESTORE LOG [StackExchange.Bicycles.Meta] FROM DISK = '\\ny-back01\backups\SQL\_Trans\SENetwork_AG\StackExchange.Bicycles.Meta\StackExchange.Bicycles.Meta_LOG_20160217_033201.trn' WITH NORECOVERY;

-- Re-join database to availability group
ALTER DATABASE [StackExchange.Bicycles.Meta] SET HADR AVAILABILITY GROUP = [SENetwork_AG];
ALTER DATABASE [StackExchange.Bicycles.Meta] SET HADR RESUME;


Afer running the above on the secondary server for both databases they were able to start synchronizing again.

UPDATE: We had a similar issue where after a Manual AG Failover one of the databases on the new primary replica was stuck in Not Synchronizing mode (switched to Not Synchronizing / Recovery Pending after restarting SQL Server), and the above steps worked to resolve that issue as well.