Why don't we build an interferometer between Earth and the Moon? The Event Horizon Telescope (EHT) does interferometry from an array of telescopes spread all across Earth. The data is locally stored on a hard drive and shipped to a central location, so the telescopes in the array don't need to be connected to each other in real time. Would it be possible to make an EHT-like interferometer with telescopes on the Earth and on the Moon? It seems like such a telescope would have a much higher resolution than the EHT, and would make it possible to observe in wavelengths shorter than radio. 

Of course, money is likely an issue here, but is there any technical reason why this wouldn't be feasible?
Put simply: controlling the phasing would be impossible (not just difficult).  Not only do you have to deal with a constantly changing physical distance but also with the dynamics of the atmosphere.  It's difficult enough to maintain transverse phasing across an earth-based receiver (think adaptive optics);  I can't imagine a reliable way to track the radial variation.  Without that, unless you are sticking with multi-meter wavelength radio systems, you cannot hope to phase up the receivers.