How do I transfer data and settings from one Apple Watch to another? I have an Apple Watch. I have had it for a while. I just bought a new Apple Watch. (Both are Series 1, just different sizes.)

I would like to transfer all my data and settings from my old Apple Watch to my new Apple Watch and then sell my old Apple Watch.

Is it possible to migrate my profile from one Apple Watch to another?

If so, what is the procedure?

All the search results I have found have to do with moving an Apple Watch to a new iPhone or restoring a backup to the same Apple Watch. Perhaps it is possible to restore a backup from an old Apple Watch onto a new one?
iMore provides a step-by-step set of instructions to restore an old backup onto a new watch that I have summarized below.

The first step is to unpair the old watch from your phone. As part of that, a backup of your watch's information and settings will be created on the phone.

Next, pair your new watch following the normal procedures - launch the Apple Watch app, tap Start Pairing, and align the watch in the viewfinder.

On the following screen it will state "Your Apple Watch Is Paired." From here, tap on Restore from Backup, and you should be given the option of the backup you made when unpairing the old watch. Tap on that.

You will then need to agree with the terms and conditions, as well as enter your Apple ID username and password. Tap through the remaining screens of setting options, including a passcode, and finally it will begin the process of restoring your backup, which may take several minutes. 

Once the restore is complete, you will have your new watch set up with all your settings and data.