Maybe you should see it as "what you should be mindful of"
instead of "who should be doing the mindful thinking"
. And don't worry about Anatta!
Personally, I can tell you it is a very very difficult concept to understand and apply by merely reading suttas, commentaries and articles on it. Trying to understand Anatta at this level and using such coarse techniques just doesn't work for most people and will just lead to false speculations and frustration.
Even the discourse on Anatta
is just not helpful and I suspect it was kept that way because it is not meant to be understood by just talking, reading and thinking about it.
But if you read the Dhamma-Cakkappavattana Sutta, it is peppered with things like:
"Vision arose, insight arose, discernment arose, knowledge arose, illumination arose within me..."
After The Buddha talks about the Noble Eightfold Path, "This is the middle way realized by the Tathagata that - producing vision, producing knowledge..."
So don't worry about concepts like Anatta for now. Keep practicing, be mindful, be alert and above all else, be patient. Trust that such knowledge will arise within oneself in the due course of time.