Update:
Since the original question, the white spots have also appeared on my calamint, and just the other day I noticed them on the leaves of the little thyme starts I planted out (which I grew from seed.)
Here is the white spotting on the calamint (with a fly who, I am sure, is unrelated, though pretty.)
Here is the thyme. One plant also has black "stuff" near the white spots.
I'm not really concerned, just quite curious. I've always felt herbs were proof against so many issues that other plants are vulnerable to, because of their strong scent perhaps -- but this? seems to target the strongly scented plants specifically!
Edit: I look at the backsides of these leaves and see nothing. From some appearances it looks similar to leafminers, but I'm sure it is not. White dots, not channels, and nothing showing on the leaf underside. It does kind of look like egg laying in there though. In mint and thyme? So odd.