"The rotation of Mercury was not discovered until 1965. Until then the most widely accepted theory had Mercury tidal locked to the Sun. Soviet scientists bounced radar signals off the planet’s surface in 1962 verifying that the planet rotated, but it wasn’t until scientists using the Arecibo Observatory verified the planet’s sidereal rotational period of 58.647 day."
(From: https://www.universetoday.com/14008/rotation-of-mercury/)
With clever tricks they can then get a hint of the size or other parameters of the object from the shape of the reflected signal.
I wouldn't have thought doing active radar scans of an astroid out in space was possible given that radar returns fall off as the 4th power of range. But given that power and dish size, given that these are large objects being tracked, and given that they move slowly compared to the radar cone I guess it's possible after all. But then again one of the first ballistic missile defense radars accidentally mistook the rising Moon for an attack and almost caused WWIII[2] so I guess there's precedent, even though the Moon is much larger and closer. I'm seeing references to it tracking asteroids in the Apollo group, so relatively nearby[3] and plates which are much larger. I'd guess that even the biggest asteroid in the main belt, 1 Ceres, would be too dim for it to see but I'm curious.
[1]https://en.wikipedia.org/wiki/Arecibo_Observatory
[2]https://blog.ucsusa.org/david-wright/the-moon-and-nuclear-wa...
They are in good company; like 90% of UFO sightings are just people seeing the Moon and mistaking it for something else (or rather, not immediately identifying it, leaving them with impressions like "there was a bright light following me down the road" or "there was a glowing orb behind the trees").