Wu Wei Wu nicely summarizes the teaching of Zen and Advaita on the non-existence of the ego-self:

If one seeks to rid oneself of, or even to transcend, a false self, ego, or personality, one thereby accepts as a fact the existence of such [an] entity and so-doing affirms its strangehold (a constraint can be real or imaginary–such as that of the chicken’s beak held by a chalk-line).

Why Lazarus Laughed: The Essential Doctrine, Zen–Advaita–Tantra, p. 6.

In my exposition, let me elaborate on what I’ll term “the early teaching” and then on “the later, advanced, direct teaching.”

I. The Early Teaching

The early teaching presumes the existence of samskaras–or false identifications–in order to allow the practitioner to skillfully identify all such samskaras. In my view, it can be quite helpful not just to vaguely label some experience as derivable “from ego.” Instead, clarity makes a difference: “I am powerless,” and this experience is arising on behalf of this samskara.

This early teaching is a fudge, yes, in order for there to be an opening to a taxonomic inquiry. (For more, see, e.g., this post.)

II. The Later, Advanced, Direct Teaching

The later teaching undercuts the central assumption by inviting the practitioner to ask: “Is there a samskara in the first place? Does it really exist?”

1. Does samskara X even exist?

2. That is, “Can I find a separate self (i) in the thought (I am powerless) or (ii) behind this thought?”

As you go looking for the existence (sat) of this separate self, all you’ll find is Awareness. In the case of (i), the thought merges with Awareness, thereby “becoming” nothing but Awareness. And in the case of (ii), what is “behind” thought is only ever Awareness.

III. Two Steps

Through these two steps, we can dispel the illusion of the existence of egoity (this illusion is called avidya or ajnana in Advaita) and thereby immediately grasp Wu Wei Wu’s conclusion: “That of which we need to rid ourselves, to transcend, is the false concept whereby we assume that entity’s existence” (p. 6, my emphasis). Or as Ramana Maharshi often says, “All you need to do is to get rid, right now, of the notion that you are not presently Self-realized.”

First, unknow what you aren’t. Second, know directly what you are.