  Let $D$ be a bounded homogeneous domain in $\mathbb{C}^n$. In this paper, we
study the bounded and the compact weighted composition operators mapping the
Hardy space $H^\infty(D)$ into the Bloch space of $D$. We characterize the
bounded weighted composition operators, provide operator norm estimates, and
give sufficient conditions for compactness. We prove that these conditions are
necessary in the case of the unit ball and the polydisk. We then show that if
$D$ is a bounded symmetric domain, the bounded multiplication operators from
$H^\infty(D)$ to the Bloch space of $D$ are the operators whose symbol is
bounded.
