Wind speed can be measured remotely, with varying degrees of success, using interferometry of Doppler-shifted optical spectra. Under favorable conditions, active systems using laser pulse backscatter are capable of high resolution; passive systems, which measure Doppler shifts of atmospheric emission lines in the mesosphere, have also been shown. Two-beam interferometry of Doppler-shifted absorption lines has not been previously investigated; we describe such an effort here. Even in a well-defined environment, measuring absorption line Doppler shifts requires overcoming several technical hurdles in order to obtain sensitivity to wind speeds on the order of 10 m/s. These hurdles include precise knowledge of the shape of the absorption line, tight, stable filtering, and understanding precisely how an interferometer phase should respond to a change in the absorption profile. We discuss the instrument design, a Michelson interferometer and Fabry-Perot filter, and include an analysis of how to choose the optimal optical path difference of the two beams for a given spectrum and filter. We discuss two beam interferometric measurements of emission line and absorption line Doppler shifts, and include an illustration of the effects of filtering on LIDAR Doppler interferometry. Finally, we discuss the construction and implementation of a Michelson interferometer used to measure Doppler shifts of oxygen absorption lines and present results obtained with 5 m/s wind speed measurement precision. Although the theoretical shot noise limited Doppler wind speed measurement of the system described can be less than 1 m/s, the instrument's resolution limit is dominated by residual filter instability. Application of absorption line interferometry to determine atmospheric wind speeds remains problematic.