These kinds of observations have frequently been made for lensed quasars. Quasars are fairly stochastic and have random bursts in their brightness. If you see multiple images of the same quasar, you can see one of the images get brighter and then some time later (usually of order days) you can see the other image get brighter.
By measuring the time delay and the separations between the images you can get an estimate of the Hubble constant.
You would need a gravitational lens so powerful it not only bent light, but made the light perform a complete slingshot around it so that it would be aimed back at Earth. The nearest such object is the Milky Way's black hole, and there's no way we'd be able to resolve Earth in an image coming from a) that far away and b) that close to the galactic core where so much other stuff is in the way.
But could a photon make such a journey? Yes. But you'd never be able to recognize it against everything else.
To correct myself, because I didn't actually bother to check first, there are actually black holes or probable black holes closer than the center of the galaxy. Like, a lot closer. [1] And we've seen a lensing effect from at least one of them. And moreover that one happens to be isolated from other material. [2]
Is there anything we can do deliberately to use gravitational lensing other than aiming a telescope at a part of the sky where this is happening by random chance?
Using interferometery of orbiting observatories (potentially on solar orbits, to leverage the entire diameter of the solar system), this is conceivably possible
That's pretty amazing, I'd certainly not expect more than a small time shift since it still managing to get the same light source to the same location to be detected.
The lensed light has generally traveled for billions of years, and wound up being separated by thousands of light years before the gravitational lensing. Two paths taking days apart DOES count as only a small time shift!
