Regarding the suggestions about noise cancellation: they won't work for cicadas. There are basically two approaches when working with multiple microphones: phase independent filtering, and phase-specific filtering.
For the phase independent filtering, you get the characteristics of the noise and dampen the frequencies according to the expected signal-to-noise ratio in all of the related frequencies. Since cicadas are wide-spectrum noise, you'll be hitting all of the good content as well. So it's only workable when generally you are either quite louder than the cicadas or silent. This sort of frequency-dependent noise-gating cannot help with content at quiet levels, only with full silence in a particular band, and cicadas are wide-band. So either you are above cicada level in some frequency and the gating does nothing, or you aren't, and the gating puts you down along with the cicadas.
For phase dependent filtering, you basically do the same as "echo cancellation" (which subtracts the phase-corrected microphone input expected to result from the current speaker output from the actual microphone input before sending it on, so that as little as possible from the speaker output is sent back on the line) in a phone communication does: you try subtracting a phase-corrected version of the noise from your main microphone signal. This can work with noise levels above the signal level but it requires correlated versions of noise on the mics. A single cicada in the room (or even a single responsible opening) would be a good use case.
However, cicadas distributed around the house and with their noise coming in at multiple places are producing ambient sound where you don't really have reliable phase relations. Also you need some separation of "noise mics" and "signal mics" so that the noise mics don't get too much of the useful information. The annoying part of the chirps is at a wavelength quite below one foot, so echo cancellation techniques will be quite sensitive to exact direction/correlation. Anything but direct sound will not cancel well.
So your best bet, if the neighborhood is suitable, is likely recording sessions at late night. The next best bet is passive dampening as much as you can. If you cannot bring the cicadas down to a level where they are no problem, at least you might have a chance to bring them down to a level where noise-gating will be effective, namely where you need to only rescue actual silence from the cicadas.