The spectroscopic characterization of terrestrial exoplanets over a wide spectral range from the near- to the mid-infrared will be made possible for the first time with the JWST. One challenge is that it is not known a priori whether such planets possess optically thick atmospheres or even any atmospheres altogether. However, this challenge also presents an opportunity, the potential to detect the surface of an extrasolar world. This study explores the feasibility of characterizing with the JWST the atmosphere and surface of LHS 3844b, the highest signal-to-noise rocky thermal emission target among planets that are cool enough to have nonmolten surfaces. We model the planetary emission, including the spectral signal of both the atmosphere and surface, and we explore all scenarios that are consistent with the existing Spitzer 4.5 μm measurement of LHS 3844b from Kreidberg et al. In summary, we find a range of plausible surfaces and atmospheres that are within 3σ of the observationless 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 ≳100 ppm are included. We further make predictions on the observability of surfaces and atmospheres and find that a small number (∼3) of eclipse observations should suffice to differentiate between surface and atmospheric features. We also perform a Bayesian retrieval analysis on simulated JWST data and find that the surface signal may make it harder to precisely constrain the abundance of atmospheric species and may falsely induce a weak H2O detection.