One, by whom the supreme Brahman is realized, will think he is Brahman. But what will the non-thinking person, who sees no second entity at all, think?
When an earnest seeker comes to know that Brahman, the supreme Reality, is single and omnipresent, he will be thinking and reminding himself that he is Brahman. The thought, the identification, leads him more and more to the Self within. And there comes a state when he realizes the truth of what he thinks, namely that he is Brahman, everything is Brahman, nothing besides Brahman is or can be. This is quite natural, even irresistible.
Think further. What will be the state of the Knower, for whom there is, there was, there will be nothing besides Brahman, because it is omnipresent? There will be no desire for anything, including liberation or realization, as nothing is besides Brahman. Also, no subtle action, which implies movement, vibration or thought is possible. Omnipresence will not move at all. What will such a thought-free or non-thinking person, the enlightened, do then?
Even the ‘I am Brahman’ thought, remembrance or practice becomes redundant. Does not one travelling to his destination stop his effort, when he reaches the destination? Any further walking will only dislodge him from where he should be!
The seeker practises brahma-vṛtti to realize Brahman. Once the realization is had, finding no second presence, he will not think or practise anything at all, instead be free and harmonious!