  We present results from a monitoring campaign performed with the Chandra
X-ray Observatory of the gravitationally lensed quasars RX J1131-1231 and HE
1104-1805. We detect significant X-ray variability in all images of both
quasars. The flux variability detected in image A of RX J1131-1231 is of
particular interest because of its high amplitude (a factor of ~ 20). We
interpret it as arising from microlensing since the variability is uncorrelated
with that of the other images and the X-ray flux ratios show larger changes
than the optical as we would expect for microlensing of the more compact X-ray
emission regions. The differences between the X-ray and optical flux ratios of
HE 1104-1805 are less dramatic, but there is no significant soft X-ray or dust
absorption, implying the presence of X-ray microlensing in this system as well.
Combining the X-ray data with the optical light curves we find that the X-ray
emitting region of HE 1104-1805 is compact with a half-light radius ~ 6 r_g,
where the gravitational radius is r_g = 3.6 x 10^14 cm, thus placing
significant constraints on AGN corona models. We also find that the
microlensing in HE 1104-1805 favors mass models for the lens galaxy that are
dominated by dark matter. Finally, we better characterize the massive
foreground cluster near RX J1131-1231, set limits on other sources of extended
X-ray emission, and limit the fluxes of any central odd images to be 30-50 (3
sigma) times fainter than the observed images.
