In Non-Duality teaching there is not a thinker or a doer. There is only Consciousness. Consciousness can't sing and dance, and it can't walk. But we seem to be able to move about. Let's understand this.
Consciousness is stillness beyond any sense of movement. Movement is the appearance of change within the unchanging Consciousness. We are not the body that seems to move about. So although you may say, "I'm going for a walk," it's not really you that goes walking. It's the body that walks, not Consciousness. The body and its movements are not who we are. We perceive the body and its movements. Most spiritually aware people could agree that they are not the body.
Then we have the mind. We might say, "I was thinking about what you said yesterday." But we don't do the thinking. The mind thinks, or rather thoughts appear in what we call 'the mind'. We are the Consciousness experiencing the mind, its thinking and thoughts. We perceive the thoughts appearing and the sense of identity in the mind that thinks it is thinking thoughts. We can't really define 'the mind'. It is what we call the field in which thinking, feeling, sensing and perceiving appears. Ultimately 'the mind' can be considered as a focusing of Consciousness.
It's important to remember that Consciousness is like a Singularity. It is dimensionless and formless. It doesn't go anywhere or do anything. There is nothing and nowhere other than Consciousness for it to be able to move or do anything. Consciousness is formless. If there is any form (including thought forms) or there is any movement, that's not you. That's not Consciousness. It appears in Consciousness, yes. But it is illusory in the sense that it is not a true portrayal of Consciousness. Consciousness cannot 'see' itself. So anything that is perceived cannot be it. Any appearance of forms will be imperfect compared to the Pure Perfection of Formless Consciousness.
So who thinks the thoughts? Nobody. The thoughts float by within Consciousness, as part of the mind and its conditioning. The mind forms a sense of identity, which is thought-based. This thought-based identity then thinks that it is thinking thoughts. Its all a flow of thought energy that reacts to the more dense forms of the body and the world. We can understand it like this: there is solid matter of the body that changes slowly, there is liquid matter in the body that moves around, there is gas also flowing through the body at a higher rate; then there is the less dense but more volatile flowing of the mind. The flowing of the mind is part of this system that interacts with the body and the world. The mind itself can have some fixed views and more flexible views. This is all perceived in Consciousness.
So there is nobody thinking. Thoughts flow. Consciousness witnesses this. The mind forms an attachment and identity as a thinker, believing itself to be a separate entity that can think. Truly Consciousness doesn't think. The nature of Consciousness is peace, stillness, and infinite oneness.