Restore the database files to a location with enough space to hold the entire database and its logs. I created a full system (bare metal) backup using Windows Server Backup. I recovered using the following options:
Select Recovery Type: Applications
Select Application: Exchange
Specify Recovery Options: Recover to another location
Under Confirmation, I presume you would be able to choose the database you want to restore. I only had one, so it was selected.

This will restore the database and logs to the location you specified. I suspect the process will be very similar for other backup solutions that support Exchange.

Check database State:

[PS] >eseutil /mh ‘E:\WSBRecovery\Mailbox Database 1234567890\Mailbox Database 1234567890.edb’

Filter output for the State:

[PS] >eseutil /mh ‘E:\WSBRecovery\Mailbox Database 1234567890\Mailbox Database 1234567890.edb’ | findstr “State”

Check status of the logs:

[PS] >eseutil /ml ‘E:\WSBRecovery\Mailbox Database 1234567890\E00’

Replay the logs and update the database. The /l option is the log location, and the /d is the database location.:

[PS] >eseutil /r E00 /i /l ‘E:\WSBRecovery\Mailbox Database 1234567890’ /d ‘E:\WSBRecovery\Mailbox Database 1234567890’

Check the State again. It should be Clean Shutdown now.:

[PS] >eseutil /mh ‘E:\WSBRecovery\Mailbox Database 1234567890\Mailbox Database 1234567890.edb’ | findstr “State”

If the State still say Dirty Shutdown, you can try to repair the database. However, this should be your last resort:

[PS] >eseutil /p ‘E:\WSBRecovery\Mailbox Database 1234567890\Mailbox Database 1234567890.edb’

Create a new recovery database using the restored clean database (make sure to use an empty directory for the logs):

[PS] >New-MailboxDatabase RecoveryDatabase -Server MY-EXCHG01 -Recovery:$true -EdbFilePath ‘E:\WSBRecovery\Mailbox Database 1234567890\Mailbox Database 1234567890.edb’ -LogFolderPath ‘E:\WSBRecovery\Mailbox Database 1234567890\Logs’

Mount the recovery database:

[PS] >Mount-Database RecoveryDatabase

If you have an old recovery database open, you may need to dismount the old and delete it first. Not sure if that is applicable for all Exchange versions, but this was the case in Exchange Server 2010 Standard version:

[PS] >Dismount-Database ‘MY-EXCHG01 DB01 Recovery’
[PS] >Remove-MailboxDatabase -Identity ‘MY-EXCHG01 DB01 Recovery’

Check for the mailboxes in the recovered database:

[PS] >Get-Mailboxstatistics -Database RecoveryDatabase

Restore the database to a Restore folder, so the lost messages can be copied from the Restore folder. Once the lost messages are moved out of the Restore folder, the Restore folder can be deleted.

[PS] >Restore-Mailbox -Identity “JGZ Test” -RecoveryDatabase RecoveryDatabase -RecoveryMailbox “JGZ Test” -TargetFolder Restore

If you are restoring a mailbox for one user and putting it in another users mailbox, you need to make sure you have the order right.
Identity is the mailbox you are going to be restoring to (target), and the RecoveryMailbox is the mailbox you are restoring (source).

[PS] >Restore-Mailbox -Identity “put-IT-here” -RecoveryDatabase RecoveryDatabase -RecoveryMailbox “get-IT-here” -TargetFolder Restore

If you get a message when restoring a mailbox that states that it is unable to restore because the maximum number of corrupt items has been exceeded, you just need to add the -BadItemLimit to the Restore-Mailbox command:

[PS] >Restore-Mailbox -Identity “put-IT-here” -RecoveryDatabase RecoveryDatabase -RecoveryMailbox “get-IT-here” -TargetFolder Restore -BadItemLimit “number-of-bad-items-permitted

I usually start low and increase until I get it restored. You can attempt to repair the mailbox too.