Who or what is observing I've heard it taught by various secular teachers that through mindfulness one can uncover the "true authentic self" because we realise that we are not thought but that we are awareness of thought. Please correct me if I'm mistaken but in my understanding this is not in alignment with Buddhist teachings and a misunderstanding of mindfulness.  We can discover that there is an awareness of thought, emotion etc but are we this awareness? Wouldn't Buddha say that even awareness is not self?  If we are not thought, not awareness, who or what is the observer? Is the observer just more impermanent phenomena?
The mind observes. Consciousness observes. Awareness observes. 


  I've heard it taught by various secular teachers that through mindfulness one can uncover the "true authentic self"


No. Buddhism does not teach there is a "true authentic self". 


  because we realise that we are not thought but that we are awareness
  of thought.


No. Ultimately, there is no "we". Buddhism explains there are only five aggregates, namely: 


Physical body
Feeling/sensation
Perception
Thinking/emotions
Sense consciousness



  Please correct me if I'm mistaken but in my understanding this is not in alignment with Buddhist teachings and a misunderstanding of
  mindfulness.


Correct. What various secular teachers are teaching is not in alignment with Buddhist teachings.


  We can discover that there is an awareness of thought, emotion etc
  but are we this awareness?


The mind discovers &amp; the mind is aware. Consciousness knows thought, as follows:


  Dependent on the intellect &amp; ideas there arises consciousness at the intellect. The meeting of the three is contact. 
  
  If anyone were to say, 'The intellect is the self,' that wouldn't be tenable. The arising &amp; falling away of the intellect are discerned.
  And when its arising &amp; falling away are discerned, it would follow
  that 'My self arises &amp; falls away.' That's why it wouldn't be tenable
  if anyone were to say, 'The intellect is the self.' So the intellect
  is not-self. If anyone were to say, 'Ideas are the self,' that
  wouldn't be tenable... Thus the intellect is not-self and ideas are
  not-self. If anyone were to say, 'Consciousness at the intellect is
  the self,' that wouldn't be tenable... Thus the intellect is not-self,
  ideas are not-self, consciousness at the intellect is not-self. If
  anyone were to say, 'Contact at the intellect is the self,' that
  wouldn't be tenable... Thus the intellect is not-self, ideas are
  not-self, consciousness at the intellect is not-self, contact at the
  intellect is not-self.
  
  MN 148
  
  
  
  'Consciousness, consciousness': Thus is it said. To what extent, friend, is it said to be 'consciousness'?
  
  'It cognizes, it cognizes': Thus, friend, it is said to be 'consciousness.' And what does it cognize? It cognizes 'pleasant.' It
  cognizes 'painful.' It cognizes 'neither painful nor pleasant.' 'It
  cognizes, it cognizes': Thus it is said to be 'consciousness.'
  
  Discernment (wisdom) &amp; consciousness, friend: Are these qualities conjoined or disjoined? Is it possible, having separated them one from
  the other, to delineate the difference between them?
  
  Discernment (wisdom) &amp; consciousness are conjoined, friend, not disjoined. It's not possible, having separated them one from the other, to
  delineate the difference between them. For what one discerns, that one
  cognizes. What one cognizes, that one discerns. Therefore these
  qualities are conjoined, not disjoined, and it is not possible, having
  separated them one from another, to delineate the difference between
  them.
  
  MN 43





  Wouldn't Buddha say that even awareness is not self?


Yes. 


  If we are not thought, not awareness, who or what is the observer?


The mind is the observer; awareness is the observer; but "we" is not the observer.


  Is the observer just more impermanent phenomena?


Yes. Refer to SN 12.61


  It would be better for the uninstructed run-of-the-mill person to hold to the body composed of the four great elements, rather than the
  mind, as the self. Why is that? Because this body composed of the four
  great elements is seen standing for a year, two years, three, four,
  five, ten, twenty, thirty, forty, fifty, a hundred years or more. But
  what's called 'mind,' 'intellect' or 'consciousness' by day and by
  night arises as one thing and ceases as another. Just as a monkey,
  swinging through a forest wilderness, grabs a branch. Letting go of
  it, it grabs another branch. Letting go of that, it grabs another one.
  Letting go of that, it grabs another one. In the same way, what's
  called 'mind,' 'intellect' or 'consciousness' by day and by night
  arises as one thing and ceases as another.
  
  SN 12.61