  The spectroscopic characterization of terrestrial exoplanets will be made
possible for the first time with JWST. One challenge to characterizing such
planets is that it is not known a priori whether they possess optically thick
atmospheres or even any atmospheres altogether. But this challenge also
presents an opportunity - the potential to detect the surface of an extrasolar
world. This study explores the feasibility of characterizing the atmosphere and
surface of a terrestrial exoplanet with JWST, taking LHS 3844b as a test case
because it is the highest signal-to-noise rocky thermal emission target among
planets that are cool enough to have non-molten surfaces. We model the
planetary emission, including the spectral signal of both atmosphere and
surface, and we explore all scenarios that are consistent with the existing
Spitzer 4.5 $\mu$m measurement of LHS 3844b from Kreidberg et al. (2019). In
summary, we find a range of plausible surfaces and atmospheres that are within
3 $\sigma$ of the observation - less reflective metal-rich, iron oxidized and
basaltic compositions are allowed, and atmospheres are restricted to a maximum
thickness of 1 bar, if near-infrared absorbers at $\gtrsim$ 100 ppm are
included. We further make predictions on the observability of surfaces and
atmospheres, perform a Bayesian retrieval analysis on simulated JWST data and
find that a small number, ~3, of eclipse observations should suffice to
differentiate between surface and atmospheric features. However, the surface
signal may make it harder to place precise constraints on the abundance of
atmospheric species and may even falsely induce a weak H$_2$O detection.
