How to create single-click/-keystroke shortcut to start recording on Mac OS?
I wrote two scripts for you to do this in QuickTime Player:

Script to start recording:

tell application "QuickTime Player"
    activate
    start (new audio recording)
end tell


Script to end recording:

Edit (21/12/2018): this script uses UI scripting, which is not recommended, but seems the best option for now. If you have a better solution, that addresses the issues that are stated in the comments, please edit this answer.

tell application "QuickTime Player"
    activate
end tell

tell application "System Events"
    tell process "QuickTime Player"
        tell window 1
            click (first button where its accessibility description = "stop recording")
        end tell
        click menu item "Save…" of menu "File" of menu bar 1
    end tell
end tell


After the second script a dialog will open, which asks you to enter a file name and save location.

To make this work, you have to copy both scripts and paste them into two separate files in Script Editor (default installed on your Mac). Save the files and follow this guide to assign shortcuts to them.

Update 21/12/2018: I updated the start script with user3439894's suggestion. I kept the stop script the same, because his stop script suggestion doesn't seem to work. (A window with 'Finishing Recording' appears, but keeps loading)