In this informative and concise set of notes on vanishing cycles by Donu Arapura, it is stated that the theory of vanishing cycles ports nicely to the etale world if the role of the disk is replaced by that of the spectrum of a Henselian DVR.
Let $k$ be algebraically closed. My intuition tells me that both a complex disk of an arbitrarily small radius and the spectrum of a Henselian DVR over $k$ are something like a local universal cover as far as the respective cohomology theories are concerned. Really, the disk is contractible in the analytic topology, so it is its own universal cover. Is my intuition leading me astray when I think that taking the Henselization of $\mathbb{A}^1_k$ at the origin is something like choosing a separable and/or algebraic closure of $k(x)$, the local ring at the origin?
Here is my motivation: suppose I want to compute the etale cohomology of some nonsingular affine hypersurface $X$ over $k$ algebraically closed of arbitrary characteristic. When is it possible to take something like a global Henselization of $X$ in some meaningful way in order to obtain something like a cofinal etale cover of $X$ sufficient for computing its etale cohomology? I could imagine there is a compatibility condition pertaining to an absolute Galois action which could render this approach intractable. Also perhaps the situation of my question is much better understood when $X$ is a curve. If it is not always possible for curves, or for general nonsingular hypersurfaces, why not/what does the obstruction look like? If it is, then what if $X$ is singular?
Finally, if my intuition about "local universal covers" is incorrect, is there any interesting way in which a disk in the analytic topology is like the spectrum of a Henselian DVR in the etale topology?