A consequence of the Riemannian Goldberg-Sachs theorem is the fact that the Kähler form of an Einstein Hermitian surface is an eigenform of the curvature operator. Referring to this property as $*$-Einstein condition we obtain a complete classification of the compact locally homogeneous $*$-Einstein Hermitian surfaces. We also provide large families of non-homogeneous $*$-Einstein (but non-Einstein) Hermitian metrics on $ℂ{\mathbb{P}}^2♯{\overline{ℂ\mathbb{P}}}^2$, $ℂ{\mathbb{P}}^1\times ℂ{\mathbb{P}}^1$, and on the product surface $X\times Y$ of two curves $X$ and $Y$ whose genuses are greater...