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)×106 Ms. For ASASSN-14li, the SMBH mass is 10(+1, -7)×106 Ms and the spin is 0.998(+0, -0.7). 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 0.998(+0, -0.15). The mass accretion rate of ASASSN-14li decays slowly, as t-1.1, perhaps due to inefficient debris circularization. Over 1100 days, its SMBH has accreted mass ~ 0.17 Ms, implying a progenitor star mass of > 0.34 Ms, 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.