Do Buddhas think?

https://www.lionsroar.com/ask-the-teachers-18/

This is an interesting piece, because of the very different answers that various Buddhist teachers give to the question "do Buddhas think?"

One teacher:

As I understand it, the awakened mind includes thinking; it’s just that the thinking is necessary, functional, and discerning rather than indulgent, unnecessary, and addictive. Without attachment to thinking, silence and peace are available. In not grasping after thoughts and taking them to be me and mine, there is freedom instead of bondage.

Buddhas plan, but don’t engage in worry. They make decisions but are not swayed by self-centered emotions. Buddhas are immeasurably creative but not interested in fantasy. Buddhas think but are not caught up in their thoughts, and do not mistake their thoughts to be who they are. They are present in the midst of thoughts arising, and use thinking as a way to benefit all beings.

This is a position I've heard many times: the thinking is reduced to the essentials. Direct! Necessary! Functional! Sounds very... Zen. Obviously you couldn't function without any thought, right? How would you know what to do?

The second teacher agrees:

The short answer is, yes, buddhas do think. However, I have a bumper sticker on my car that says, “Don’t believe everything you think,” because so often we identify with our thoughts and set up a self there.
...
“Who is it that notices these thoughts as wholesome or unwholesome and responds for the benefit of all beings?” This is an example of how a buddha or bodhisattva thinks. 

But I think these are wrong. This has all been mapped out by the Dzogchen tradition. Tulku Urgyen Rinpoche (TUR):

What this really means is that we should repeatedly look into what thinks. We should recognize the absence or emptiness of this thinker over and over again, until finally the power of deluded thinking weakens, until it is totally gone without a trace.

But this is not how a Buddha practices; this is how a beginner practices. Recall from the first quote:

They are present in the midst of thoughts arising, and use thinking as a way to benefit all beings.

Dzogchen talks about three stages of capacity:

Within this method of [cherdrol], "liberation through bare attention," there exists a minimal mental activity; we turn our attention to the thought as soon as it arises and recognize it as a thought. ... Although this may have become a largely automatic process, there still exists a small time gap between the arising of the thought, the becoming aware of its presence, and the recognition of it as a thought, on the one hand, and its dissolving again, on the other hand. For the beginner this practice is appropriate, but later it will become a fault if it is not transcended. 

At the next stage, the thought disintegrates as soon as it arises. Patrul Rinpoche illustrates this process with the example of drawing pictures on the surface of the water. The picture disintegrates as soon as it is drawn.
... 

With the third stage, even the distinction between arising and liberating or dissolving is transcended. ... Thoughts liberate as they arise; their very arising is the process of their liberating. This is the method proper to Dzogchen, and all else is but preparation.

When thoughts liberate simultaneously with their arising, they are no longer "thoughts" at all. This is very hard for the mind to understand. TUR again:

As conceptual thinking diminishes and finally vanishes, what is left to cause us to wander in samsara? The very basis for samsaric existence is none other than conceptual thinking.

This is what the third teacher (also a Dzogchen master) indicates:

By definition, a buddha—one who is free of the suffering of existence—does not have a grasping mind. Since a buddha does not grasp, a buddha does not have thoughts. 

A buddha’s perception is pure awareness, or rigpa, which is not a product of the moving, thinking mind, but is direct perception. Thought can never experience the true nature of mind directly, so in Dzogchen, thought is not encouraged since it will not liberate us from suffering. And while conventionally we could agree that the thought to benefit another is preferable to the thought of jealousy, in order to achieve full realization one needs even to be free of positive thoughts because of their involvement with the grasping mind.

The hardest part about getting here is that it seems very clearly impossible to function without thoughts. Therefore, we set up this idea that in the awakened state, we will have only the necessary thoughts. But this belief ultimately prevents us from seeing what lies completely beyond thought.