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.
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