I recently picked up a 627 PC 5" that the previous owner had worked over by a well known smith. A stainless trigger and stainless DAO hammer and a competition match job was performed. The work was done in February of this year and I purchased it about 2 weeks ago, so it hasn't seen a whole lot of use since the work was performed.

Now on this my question, when Im shooting and I'm letting the trigger reset, it feels as if it has reset full as I can feel the 'click'. But if I try to pull the trigger again it is locked, then if i let off the trigger another little bit, I can feel a second 'click' and now the trigger is fully reset.

Is this double click/Engage normal in a DAO conversion or competition trigger?

Yes, if you want it to go bang again you have to let the trigger "fully reset".
Letting the trigger less than fully return we call "short stroking" and it doesn't work.

Between the hand returning across the ratchet tooth and the mating surfaces of the trigger lip and cylinder stop sliding across each other, there are three audible clicks on a S&W revolver action's reset -- you need to let all of them happen to properly cycle the action and set-up for the next shot.

Both the above answers are correct, but incomplete. If you are releasing the trigger fully and it doesn't completely reset then either a too light rebound spring has been installed or the factory spring has been cut too much. This is not at all uncommon on a gun that has had a "trigger job" done!

Thanks for the replies.

Rebound spring is to light, trigger not traveling all the way forward.