Can single-cell organisms have through-holes? (Are donut-cells a thing?)
Yeah, holes seem to happen in leptomyxa:

L[eptomyxa] reticulata is an edaphic protozoan and a predator of bacteria and amoebae. [...] The plasmodium is very polymorphic; its size varies between 50 .mu.m and 1 cm. It generally spreads in the form of a very thin transparent protoplasmic layer. It may have the shape of a full blade, sometimes perforated with holes, or of a network; the pseudopodial zones with active locomotion present a typical fan-shaped contour.

There are some hi-res images from other sources showing the same



Image source: http://penard.de/Amoebozoa/Tubulinea/Leptomyxida/
It seems that what happens is that branches &quot;merge back&quot; after touching &quot;at the tips&quot; leaving holes in such organisms, but that's just speculation on my behalf. I haven't (yet) found a source discussing the dynamics of how these holes get formed.
And a couple more images from  a different source--a broader review of leptomyxidae.


[Above] Light microscopy of Leptomyxa arborea n. sp. and L. variabilis n. sp. (by S.
Geisen and C. Burberg) on the plastic bottom of a Petri dish (Phaco). A-F: Leptomyxa arborea,
various shapes and specimens of different size. G-M: Leptomyxa variabilis, various shapes of
trophozoites (G-K and M); cysts (L-M). Arrows indicate the most pronounced bunches of
adhesive filaments; arrowheads indicate pore like structures in cysts. Scale bar is 100 μm in A-E
50 μm in F-K and 25 μm in L-M.



[Above] Isolate of Leptomyxa sp. (by morphology of trophozoite indistinguishable from
L. reticulata) observed in the sample from Valamo Island, North-West Russia (by A.Smirnov).
A: usual appearance of a trophozoite. B-D: subsequent stages of contraction of the cell on the
object slide under the bright light beam. Figure D shows pseudopodia very similar to those or
Rhizamoeba; polypodial cell like this one can move relatively rapidly for some distance.
Molecualar data from this strain were not obtained. Scale bar is 50 μm.

By the way, the &quot;branches&quot; are more or less &quot;legs&quot;/&quot;limbs&quot; used in locomotion; but also this species is multinucleate:

Leptomyxa reticulata is the type species of the genus Leptomyxa. It is a
multinucleate, expanded organism with plasmodia-like organization of its ramose body,
possessing a simple life cycle [...] Moving amoebae usually form adhesive uroidal filaments along the posterior end of the body or an uroidal structure consisting of a pack(s) of such filaments (the latter often is seen when the locomotive cell adopts a shape of
an irregular ramose triangle, with the base directed anteriorly).

A more diagrammatic explanation:


Main morphological traits of leptomyxid amoebae. A-E: Leptomyxa cf. reticulata
(Valamo Island strain by AS). A: expanded, plasmodium-like ramose cell, slowly moving on the
substrate. This is the “textbook” appearance of Leptomyxa. B – more compact, faster moving and
actively feeding cell. C: Disturbed cell showing compact lobose pseudopodia. Leptomyxa accepts
shapes like this under certain influence (light, temperature, etc. in order to relocate rapidly). D:
solitary cyst with double wall. E: several cysts covered with the shared outer cyst wall (D-E are
drawn based on photos by Pussard and Pons 1976a)

(So yes, creepin' amoeba with holes and &quot;limbs&quot;!)
A more interesting question at this point is whether there are any (naturally occurring) single-nucleus examples of &quot;cells with holes&quot;.