  We develop a new model for X-ray emission from tidal disruption events
(TDEs), applying stationary general relativistic ``slim disk'' accretion
solutions to supermassive black holes (SMBHs) and then ray-tracing the photon
trajectories from the image plane to the disk surface, including gravitational
redshift, Doppler, and lensing effects self-consistently. We simultaneously and
successfully fit the multi-epoch XMM-Newton X-ray spectra for two TDEs:
ASASSN-14li and ASASSN-15oi. We test explanations for the observed,
unexpectedly slow X-ray brightening of ASASSN-15oi, including delayed disk
formation and variable obscuration by a reprocessing layer. We propose a new
mechanism that better fits the data: a ``Slimming Disk'' scenario in which
accretion onto an edge-on disk slows, reducing the disk height and exposing
more X-rays from the inner disk to the sightline over time.For ASASSN-15oi, we
constrain the SMBH mass to $4.0^{+2.5}_{-3.1} \times 10^6M_\odot$. For
ASASSN-14li, the SMBH mass is $10^{+1}_{-7}\times 10^6M_\odot$ and the spin is
$>0.3$. For both TDEs, our fitted masses are consistent with independent
estimates; for ASASSN-14li, application of the external mass constraint narrows
our spin constraint to $>0.85$. The mass accretion rate of ASASSN-14li decays
slowly, as $\propto t^{-1.1}$, perhaps due to inefficient debris
circularization. Over $\approx$1100 days, its SMBH has accreted $\Delta M
\approx 0.17 M_\odot$, implying a progenitor star mass of $> 0.34 M_\odot$,
i.e., no ``missing energy problem.'' For both TDEs, the hydrogen column density
declines to the host galaxy plus Milky Way value after a few hundred days,
suggesting a characteristic timescale for the depletion or removal of obscuring
gas.
