  The solution of the Calabi Conjecture by Yau implies that every K\"ahler
Calabi-Yau manifold $X$ admits a metric with holonomy contained in
$\textrm{SU}(n)$, and that these metrics are parametrized by the positive cone
in $H^{1,1}(X,\mathbb{R})$. In this work we give evidence of an extension of
Yau's theorem to non-K\"ahler manifolds, where $X$ is replaced by a compact
complex manifold with vanishing first Chern class endowed with a holomorphic
Courant algebroid $Q$ of Bott-Chern type. The equations that define our notion
of best metric correspond to a mild generalization of the Hull-Strominger
system, whereas the role of $H^{1,1}(X,\mathbb{R})$ is played by an affine
space of 'Aeppli classes' naturally associated to $Q$ via Bott-Chern secondary
characteristic classes.
