Is it possible to get my headset to initiate a recording app and record my voice?
I can't say too much here, but this thing may be achieved by an app called Tasker. Many users say it god of android. To function on every aspect of requirement, it needs root access. In your case, it does not require root access.

You can use its capability to achieve that certainly. This is wiki page of that app.

Edit :
As OP said that he has hands on Headset Button Controller. This app is also supports tasks created by Tasker app. 
Steps to Achieve this ->


  
  Open Tasker App. Go to Tasks tab. Click Plus(+) to add a task Record. Now edit the task to get it function what you want. 
  Click +, then click on Media , then Record Audio.
  Give the file name of your recorded file. Select your source ( In this case Microphone). You can select format on your choice.
  Now link this task with that app. Open app. Click on one of interaction that you want (Single click is your choice in question). 
  Now choose Tasker task and select the task (Record) as you created in Tasker app.
  You are good to go. Try it.
  


I tested it by myself. So, it must work. On more out of Tasker, use WiKi as posted above.