Monterey upgrade broke my external ssd boot

So the external drive is not actually broken in the sense it does not functioning at all, it just does not work as the Startup Disk from a cold boot.

That is problematic and dependent on a number of things including how the drive is physically connected to the computer.

Plus the type of SSD Eternal Drive.

Have seen a number of uses having this type of issue and each External Drive has been different.

Somewhere around Monterey 12.3 to 12.3.1 and again from 12.3.1 to 12.4 this type of Boot from External tie has become more and more of an ongoing issue.

Since the data is still there, would suggest lifting all the Important Data off the drive and onto another safe and secure location.

Then, Reformat the Entire External SSD and run the Monterey Installer back to that external drive with finger crossed if will work.

An aside, if booting to external Drive, further suggest having ONLY the external SSD attached to the computer. No Other externals attached.

If the User was attempting to Install Monterey Directly to the External Drive without First installing Monterey on the actual computer >> read on

There are reports appearing on the ASC Forums regarding failed installation of Monterey on older Qualifying Apple Computers. In most cases this is related to having a NON Apple Original Internal Drive ?  

In this instance, the External SSD drive being the Destination for Monterey

The EFI Boot ROM included in Monterey is verifying for Original Apple Drive and if not found , may Balk or fail to upgrade to Monterey. 

Reports indicate the EFI BOOT ROM is Required to Flash the Computer and is a  requirement for a Successful Monterey Installation.

The possible solution is, Install Monterey to the Internal Drive First.

Then it should be able to Install Monterey to the External SSD drive.

I've had better luck from the Safe Mode because I evidently have some 3rd party software that interferes with the installation so try the following:

#1 - boot into Safe Mode according to Start up your Mac in safe mode - Apple Support.

NOTE: Safe Mode boot can take up to 3-5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

#2 - go to the System/Software Update preference pane and begin the download.

#3 - run the installer from the Safe Mode.

Honestly, do not know how Monterey would install to an External Drive BEFORE the Internal Drive.

Without the EFI BOOT ROM being Flashed to the Computer, not the drive itself, I can not explain how you achieved going Directly to External First and later to the Internal Drive.

There are Technically some non approved by Apple ways but will not continue on that path

As for recovering the External Drive Data - honestly and not being smart - Professional Recovery Company does some to mind.

It is going to be dependent on exactly what Data is being lifted from the drive.

Applications, Kernel Extension and Drives would need to be reinstalled directly from the Developer.

Personal data as in documents, videos, music, pictures etc that can be moved back to the newly reformatted and re-installation drive.