So, it's possible to engage in meditation and notice that sometimes the mind is noisy, or, that one can catch themselves immersed in a train of thought a few minutes after having formed the intention to maintain attention on attention. In the extreme case, one might describe the situation of not wanting to think, but thoughts happening anyway.
That extreme case is clearly split-mind: if the individual really didn't want to think, then, they wouldn't. Noone really has two different personalities operating independently within the confines of their individuated mind, it just might seem otherwise to them. Meditative techniques that can lead to a temporary mental silence may or may not correlate with someone realizing this point about the split-mind. All the debates about this issue on the forums illustrate that this depends on the individual.
The question about "who or what stops the thinking?" is clearly just a form of the existential question, and is very similar to "what is the source of thought?". This can be pointed to in either personal or cosmic terms.