If, during the course of an investigator's turn, the clock strikes midnight twice, do we resolve two midnights after that turn? Or just one?

I seem to remember a recent ruling that said that for each Midnight strike beyond the first, simply add a doom token. Julia should be on to give more insight.

Tibs is right. In Gates of Arkham, it was added a rule saying that for each Midnight after the first, a doom token is added to the doom track instead of resolving the 2nd, 3rd... midnight.

Even if this rule was hosted in the Gates rulebook, it was also used for Omens of Ice, and it's the general rule of the thumb to follow whenever (so, whatever version of Elder Sign you're playing) 2+ Midnights are to be resolved in the same Clock phase.

It's a particularly good rule because that way you don't have impotent midnight effects that were supposed to last for "until the next midnight" or however it's phrased.