We've had some good threads of Soviet Lunar landing getting there first, that did not require such an accident to strike Apollo 11--in other words, it beat Apollo 11's OTL schedule.
The key in the one I am thinking of, named Red Star, was the Kremlin committing to a less ambitious target for N1, at an earlier date, and committing to actually going forward with this scaled back N1 earlier. Basically they omit the center six engines (which only were supposed to burn 31 seconds!) and accept a lower LEO payload--as originally conceived by Korolev's design team, N1 was supposed to be a heavy space station component launcher, not a moonship.
By that TL's diagnosis, which I think was reasonable, N1 of OTL was doomed because it was overambitious and also committed to too late. Not being able to do much to stretch the already quite gigantic stack, Mishin, Korolev's successor, pushed to lighten it instead, in order to squeeze in enough margin for a lunar Soyuz/LK stack pretty analogous to the Apollo CSM/LM stack in some ways--save that the upper stage of the N-1 would serve to brake the pair into LLO, and then IIRC have propellant left to push the Soyuz alone back toward Earth.
There is every reason to question whether LK was adequate to the job, being slashed in size and requiring a crasher stage to descend. The process of lightening the N-1 enough to enable it to push even this shrunken stack to TLI was fatal.
Now suppose we have a more robust but heavier N-1 of slightly reduced maximum thrust, which cannot by itself put even the dangerously ultralight Soyuz-LK combination into orbit along with the upper stages needed for TLI.
Here comes the clever bit--instead, the program went in the ATL story to two N-1 launches. The first one would place an LK and crasher stage into LLO, to wait there, for a second N-1 launch enabling a crewed Soyuz to proceed independently to the Moon, there to enter LLO and rendezvous with the LK.
In this way, both the Soyuz and LK, along with their respective propelling stages, can be more massive, while the N1 can be a bit simpler and more robust. To make it to the Lunar landing before Apollo 11 it also needs to be authorized much earlier, but perhaps it would work to have the space station launcher version approved quite early in the '60s; perhaps a POD could be that Korolev consolidates his position as the master of the Soviet space program with fewer rivals, and the regime commits to the "N" system as the immediate successor to the R-7 upgrades that were Vostok and Soyuz rockets, with a plan to develop a large space station from components launched on the early draft version of N1.
It should be noted, N1 was part of a larger scheme, in which a smaller rocket that essentially would be N1 minus its first stage would take the place of roughly the Proton rocket OTL. The advantage being, it would have been possible to first develop this smaller rocket, typically described as N2, testing out all the upper stack of N1, before developing the supersized N1 first stage.
By accepting that N1 could not be stretched to match a Saturn V type performance, but SpaceX style relying on the economies going with production of many instances of the same engine in large numbers (N1 would use 30 engines, in the ATL reduced or rather with designers refusing to raise it beyond 24, on the first stage, IIRC 8 of essentially the same engine on the second stage (a variant with longer vacuum optimized nozzle) and one of those engines again on the third stage) I think it just might have been possible, as in the TL, for the Soviets to get a one cosmonaut LK down just before Apollo 11 was possible.
