Do any known exoplanetary/solar bodies have "annular" eclipses similar to Earth's? This was somewhat inspired by this question but is a bit different: I am interested in if there are any known systems in which the angular size of the eclipsing object is almost the same as the angular size of the object being eclipsed from the surface of a planet, as is the case for the Moon and the Sun as viewed from Earth. This is the only way that one can get "annular" eclipses, or eclipses in which the host star's corona is clearly visible. This could be the case for some moons in the solar system (which requires that they eclipse at all), and at least Phobos in particular seems to have a similar angular size to the Sun as viewed from Mars,



but it could also be true for some known (or candidate) multiple-planet exoplanetary systems in which the periods of the two planets are not too different. Probably a perusal of the public Kepler candidate sample would be helpful here.
This will depend the position of the observer and (obviously) the relative sizes of the star and the eclipsing body.

For an intelligent observer standing on the surface of a planet the most obvious and likely candidate for an eclipsing body would be a moon of that planet. The nearer the planet is to the sun the larger that moon has to be, and the further the planet is away from the star the smaller the moon can be.

Finding planets around stars is hard enough, finding moons that will be orbiting quite closely around those planets across interstellar distances is even harder (I'm not going to say impossible, but it's pretty close) with current technology, so at the moment the answer is "None, that we know of".

Also don't forget that, according to current theories, the moon was formed when a large Mars sized body collided with the nascent Earth, the chances of another Earth/Moon type system is probably remote. This means that most natural satellites are likely to be Phobos sized bodies. Therefore to get an eclipse the planet would have to be further out from the star (so the star looks smaller) and that would tend to put it on the edge of the "Goldilocks" zone, making the likelihood of there being an intelligent observer somewhat rare.