In conversations with the Rabbi’s students, I often ask whether there is an existential/experiential dimension in the Rabbi’s thought.
I’d be glad for some clarification: can one say that the Rabbi has no existential doctrine, only a desire for consistency and precision in the arguments of faith? Or perhaps: is the Rabbi an incurable follower of the Chazon Ish and Brisk…

There is a common mistake people make, as though in Brisk there is no existential experience. And that is why they are also puzzled by Rabbi Soloveitchik, who expresses powerful experiences together with being a Brisker.

But that is not so, for two reasons:
First, in Brisk too there are experiences, but they do not determine what one does or thinks. That is determined by the mind, not by the heart.
Second, Brisk-ness itself, and the dry Lithuanian rationality, is a very powerful experience. Something like the aesthetic religious experiences Einstein describes in the face of nature and its laws. Compare Rabbi Joseph Dov Soloveitchik’s descriptions in Halakhic Man, where he stands before a spring or a sunset and sees before him the laws of one who immersed that day. Again, some people make the mistake of thinking there is no experience here, but that is not so. The experience is bound up with intellectual ideas. And it requires further inquiry whether this is an experience in the same sense as an emotional one, or perhaps something a bit different (and also located in a different part of the soul. That is, does the experiential-emotional part of a person experience the understanding and grasp of the intellect, or is the experience here itself an intellectual effect?).

I will just note that, in my humble opinion, with the Chazon Ish this appears differently than in Brisk, from several angles. Just for example: the Chazon Ish experiences the angel Sekheliel (intuition, synthetic intellect), whereas in Brisk they experience recursive understanding (the logical-juridical-analytical structure). Again, one can ask whether this is the same experiential center awakened from two different directions/sources, or whether these are two different kinds of experiences. And it seems to me that this depends on the previous conceptual inquiry (for if there they are two different kinds, the same would apply here, and vice versa), but this is not the place to elaborate.

As for what you asked about me, it seems to me that in my own small case the second type of experience exists mainly, and a little of the first. And in the second type of experience I have merited to taste from both the Chazon Ish and Brisk.
I will add that the Chazon Ish is the more mature one, while in Brisk it is a more childlike and naive manifestation. And that is also how these things appeared in me (and this is my own existentialism, as I hinted in “Two Wagons” for those in the know).

Study this very carefully, because there are very deep things here, from the furnace of the soul and of cognition, and the heart cannot fully reveal them to the mouth.

Can the Rabbi help me make some order of this—what is existentialism בכלל? And is it not rational?

That’s too broad a question, and I’m also not an expert on the matter (and in my humble opinion, nobody really is).
Briefly, I would say that existentialism is a mode of thought hewn from the thinker’s life and subjective insights (his existence) rather than from his cold, objective intellect.
For example, Kierkegaard, who is regarded as the founder of existentialism (at least the religious kind), spoke of a person’s three-stage maturation through three stages, with different forms of relation and thought: the aesthetic, the ethical, and the religious. He derived this process from what happened to him in his personal life.
In such thought there is a tension between two elements: 1. It is not based on objective arguments. 2. It is not completely subjective either. There are insights here with objective value; otherwise other people would not find value and meaning in them. So there are insights hewn from the subjective, but found to have significance by many on the objective plane.
Now you can understand that when someone does this properly, it can definitely be philosophy of value—but only because it is not really existentialist. Existence serves as a source from which intuitive insights are drawn, but overall this is intuition-based philosophy, like any other philosophy. By contrast, if it is done in a subjective and vague way, as in many cases, then it becomes a branch of psychology or just meaningless subjective verbiage.
Therefore, in my view, there is really no such creature as existentialist philosophy. Existence is the source of inspiration (parallel to what in philosophy of science is called the context of discovery), and the source of inspiration for a philosophical doctrine or system is irrelevant in judging it. The philosophical product is judged on its own merits like any other philosophical doctrine (similar to the context of justification).