What can be said about the Fourier expansion of compositions of Boolean functions, if anything?
If $f,g$ are two $\pm1$-valued functions and $f \lor g$ is their maximum (this is the standard notation for maximum; $\land$ stands for minimum), then
$$
f \lor g = \frac{1 + f + g - fg}{2}.
$$
Since $\chi_S \chi_T = \chi_{S \Delta T}$, this shows that
$$
\chi_S \lor \chi_T = \frac{1}{2} + \frac{\chi_S}{2} + \frac{\chi_T}{2} - \frac{\chi_{S\Delta T}}{2}.
$$
You can read the Fourier expansion from this (unless $S = T$, in which case $\chi_S \lor \chi_S = \chi_S$).