  The idea of using Dirac cohomology to study branching laws was initiated by
Huang, Pandzi\'c and Zhu in 2013 [HPZ]. One of their results says that the
Dirac cohomology of $\pi$ completely determines $\pi|_{K}$, where $\pi$ is any
irreducible unitarizable highest weight $(\mathfrak{g}, K)$ module. This paper
aims to develop this idea for the exceptional Lie groups $E_{6(-14)}$ and
$E_{7(-25)}$: we recover the $K$-spectrum of the Wallach modules from their
Dirac cohomology.
