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.