We present new evidence of X-ray absorption variability on time scales from a few hours to a few days for several nearby bright AGNs. The observed NH variations imply that the X-ray absorber is made of clouds eclipsing the X-ray source with velocities in excess of 103 km s−1, and densities, sizes and distances from the central black hole typical of BLR clouds. We conclude that the variable X-ray absorption is due to the same clouds emitting the broad emission lines in the optical/UV. We then concentrate on the two highest signal-to-noise spectra of eclipses, discovered in two long observations of NGC 1365 and Mrk 766, and we show that the obscuring clouds have a cometary shape, with a high density head followed by a tail with decreasing NH. Our results show that X-ray time resolved spectroscopy can be a powerful way to directly measure the physical and geometrical properties of BLR clouds.