  Let $\mathcal E$ denote the set of all unital entanglement breaking (UEB)
linear maps defined on an operator system $\mathcal S \subset M_d$ and, mapping
into $M_n$. As it turns out, the set $\mathcal E$ is not only convex in the
classical sense but also in a quantum sense, namely it is $C^*$-convex. The
main objective of this article is to describe the $C^*$-extreme points of this
set $\mathcal E$. By observing that every EB map defined on the operator system
$\mathcal S$ dilates to a positive map with commutative range and also extends
to an EB map on $M_d$, we show that the $C^*$-extreme points of the set
$\mathcal E$ are precisely the UEB maps that are maximal in the sense of
Arveson (\cite{A} and \cite{A69}) and that they are also exactly the linear
extreme points of the set $\mathcal E$ with commutative range. We also
determine their explicit structure, thereby obtaining operator system
generalizations of the analogous structure theorem and the Krein-Milman type
theorem given in \cite{BDMS}. As a consequence, we show that $C^*$-extreme
(UEB) maps in $\mathcal E$ extend to $C^*$-extreme UEB maps on the full
algebra. Finally, we obtain an improved version of the main result in
\cite{BDMS}, which contains various characterizations of $C^*$-extreme UEB maps
between the algebras $M_d$ and $M_n$.
