Javascript: How to avoid passing callback function to get return value of asynchronous function

Question

I'm building a Chrome Extension and want to get some data from local storage.

function Extension() {
  function getUser(callback) {
    chrome.storage.sync.get('user', function(user){
      callback(user);
    });
  }
  this.getUser = getUser;
});

var extension = new Extension;

Allows me to do the following elsewhere:

extension.getUser(function(user){
  console.log(user);
});

=> The user object I'm hoping for! Hooray!

But I'd rather be able to just do something like:

var userData = extension.getUser();

And receive the user object in return.

I tried changing this.getUser = getUser to

this.getUser = getUser(function(user){
  return user;
});

But that didn't work.

console.log(extension.getUser);
=> undefined

I don't believe this to be a duplicate of this answer as I'm addressing the suggestion to pass a callback function to the async request. (How do I return the response from an asynchronous call?)

To expand even further, if I modify my code to read

 this.getUser = getUser(function(user){
     console.log(user);
 });

It does log the user object to the console! So close! I seem to be successfully retrieving the object by passing a callback to the function. But my question is how can I do that in a succinct way elsewhere in my code?

===== EDIT =====

If I save a userData object as a property of the Extension function like so:

var that = this;

this.getUser = getUser(function(user){
    that.userData = user;
});

Then I can make the following clean call elsewhere in my code and it's working.

var user = extension.userData;

Is there a drawback to this or is there a cleaner way to assign userData to the Extension without the this/that method?