A binary system of a black hole and a pulsar, with the pulsar beam occasionally grazing the black hole's event horizon causing gravitational lensing, so an eclipsing binary, seems like it would a fascinating object to study if one were ever discovered (though it seems unlikely everything would line up so perfectly for us).
Perhaps lauch interstellar space probes to get to the correct vantage point. Unfortunately, a probe's data might take hundreds or thousands of years to transmit to earth.
Even more fancifully, construct such a system to study: astroengineering.
In the absence of being able to study one for real, model one in simulation.
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