Can I make macOS Sierra not notice when I turn off my monitor?
I suspect the best way to manage your situation is via some of your System Preference settings. 

More specifically, try the following and let me know how you go:


Go to System Preferences 
Select Energy Saver
Set your Computer Sleep to Never
Set your Display Sleep to a short period
Ensure that Put hard disks to sleep when possible is unchecked
Now exit Energy Saver and select Desktop &amp; Screen Saver instead
Select the Screen Saver tab
Ensure that the Start after setting at bottom left is set to Never
Exit System Preferences


Now try using your Mac Mini as you want, but don't turn off your monitor until after it has gone to sleep. Let me know how you go.

Sleep

As an aside, is there a reason you want to physically turn your monitor off?

Your monitor uses less than 0.3W when it's in sleep mode, so you're not really saving much in terms of energy by shutting it off instead of letting it sleep. So, if you set your Energy Saver settings as above, the amount of energy consumption while your monitor sleeps is negligible and your Mac Mini can still perform what it has to.