Unless I'm missing something, I would just install El Capitan (if that's what you're trying to do) and then use the Migration Assistant app within El Capitan to restore from your Time Machine backup.
For more information on this, refer to How to move your content to a new Mac on Apple's website.
In summary though, you can use Migration Assistant to transfer data, user accounts, settings and apps from your backup drive, and you can do this manually after installing El Capitan.
Migration Assistant can also be used multiple times so you can choose a combination of what you need to transfer across (e.g. applications, user account(s), settings).
Note that while it can transfer applications and utilities, those bundled by Apple with Mac OS X/macOS (e.g. Safari, Mail, etc) are not migrated across (although some user data such as your Safari bookmarks will transfer across).
You can run Migration Assistant from whatever version of macOS you have installed. By doing this you can restore your entire system from your Time Machine backup (well, as much as is possible from a Time Machine backup). Doing this is effectively the same as using the Restore From Time Machine Backup option available in the macOS Utilities window you refer to in your question.
If you want to install El Capitan first, then download it here. You can refer to How to upgrade to OS X El Capitan on Apple's website for more info.