There's very strong evidence, particularly against the Delta variant[1]. That paper cites a 33.5% effectiveness after one dose, compared to 87.9% after two doses of the Pfizer/BioNTech vaccine.
That includes the lag time. A single dose is supposed to be about 80% effective once immunity develops. They don't know how long that will last and thus have the second shot to increase effectiveness and hopefully longevity.
It's likely that the 80% figure you mention was based on protection against "ancestral type" virus, as would have been measured in the main trials. The gap between first dose and second dose protection is larger for Delta variant than for other variants, as documented in the paper I cited.
That is not consistent with the "Vaccination status" definition in the paper I cited. The tl;dr on that is that "first dose" starts 21 days after vaccination, and "second dose" 14 days after.
If you wait 21 days after the first dose, or 14 days after the second dose. Unless everyone had their second dose 7 days after the first one you still don't compare the same thing.
We need something like, 21 days after the first dose, and split the result between people who got one dose and those who got two. We also need to make sure that the repartition of ` age since first dose` is the same for the two doses group and the single dose group.