How do I file an extension through my TurboTax desktop software? The answers that I am given are for online only.

See the TurboTax Help article How do I file an IRS tax extension? for guidance on filing your extension using the desktop version of TurboTax.

If you click the FEDERAL TAXES tab you will see an OTHER TAX SITUATIONS tab.  This will open an interview page that includes "File an extension" in the Other Tax Forms section.  

You can get an extension:

You must have a TurboTax account

1. Click on the website  Free federal tax extension is now available 
2. Select [File Now]
3. Log into your account
4. Your screen will show Get started on your federal extension
5. Follow the interview

Please contact us again with any additional questions or to provide some additional details.

the suggested method only lets you file by mail since you are using desktop.

to e-file using desktop go to this Turbotax website

https://turbotax.intuit.com/irs-tax-extensions/ 

It doesn't do states if you need them but most states have a website where you can apply online for an extension. 

This process generates the Form 4868 but does NOT eFile the extension. Still have to mail.