I would add this as a comment to the previous answer, but my starter-reputation precludes me from doing this. So, here is my refinement of Karsus Ren's response.
Compton's formula (and 1923 paper) were about x-ray scattering from free electrons. In 1928, DuMond published a paper showing that for Mo K-alpha x-rays scattered off of beryllium, in addition to the Compton shift, there was a broadening of the scattered spectrum. He (correctly) guessed that this was due to Doppler broadening. The scattering electrons were not at rest, but had a broad, finite momentum distribution. He further calculated what the broadening would look like for a few different model systems. The broadening was much larger than would be expected for either atomic electrons or a classical gas of electrons. It was much better described by a gas of free electrons that obey the Pauli exclusion principle. Since no two fermions can occupy the same quantum state, electrons in a metal occupy states with much larger momentum than would otherwise be expected (the "Fermi sea").
Note that this is not thermal motion, but is instead ground-state quantum motion. That being said, this effect is currently being used to measure temperatures in laser-shock compressed matter such as that in inertial-confinement fusion experiments, where temperatures reach well above 10$^5$ K (where the thermal broadening is measurable on top of the Pauli broadening).
Other structure in the scattering spectrum comes from processes that are not Compton scattering.