We present observations of two LMC supernova remnants (SNRs), DEM L238 and DEM L249, with the Chandra and XMM-Newton X-ray satellites. Bright central emission, surrounded by a faint shell, is present in both remnants. The central emission has an entirely thermal spectrum dominated by strong Fe L-shell lines, with the deduced Fe abundance in excess of solar and not consistent with the LMC abundance. This Fe overabundance leads to the conclusion that DEM L238 and DEM L249 are remnants of thermonuclear (Type Ia) explosions. The shell emission originates in gas swept up and heated by the blast wave. A standard Sedov analysis implies about 50 M in both swept-up shells, SNR ages between 10,000 and 15,000 yr, low (0.05 cm^-3) preshock densities, and subluminous explosions with energies of 3 × 10⁵⁰ ergs. The central Fe-rich supernova ejecta are close to collisional ionization equilibrium. Their presence is unexpected, because standard Type Ia SNR models predict faint ejecta emission with short ionization ages. Both SNRs belong to a previously unrecognized class of Type Ia SNRs characterized by bright interior emission. Denser than expected ejecta and/or a dense circumstellar medium around the progenitors are required to explain the presence of Fe-rich ejecta in these SNRs. Substantial amounts of circumstellar gas are more likely to be present in explosions of more massive Type Ia progenitors. DEM L238, DEM L249, and similar SNRs could be remnants of "prompt" Type Ia explosions with young (~100 Myr old) progenitors.