  Let $S$ be a spinor bundle of a pseudo-Euclidean vector bundle
$(E,\mathrm{g})$ of even rank. We introduce a new filtration on the algebra
$\mathcal{D}(M,S)$ of differential operators on $S$. As main property, the
associated graded algebra $\mathrm{gr}\mathcal{D}(M,S)$ is isomorphic to the
algebra $\mathcal{O}(\mathcal{M})$ of functions on $\mathcal{M}$, where
$\mathcal{M}$ is the symplectic graded manifold of degree $2$ canonically
associated to $(E,\mathrm{g})$. Accordingly, we define the Weyl quantization on
$\mathcal{M}$ as a map
$\mathcal{WQ}_\hbar:\mathcal{O}(\mathcal{M})\to\mathcal{D}(M,S)$, and prove
that $\mathcal{WQ}_\hbar$ satisfies all desired usual properties. As an
application, we obtain a bijection between Courant algebroid structures
$(E,\mathrm{g},\rho,[\cdot,\cdot])$, that are encoded by Hamiltonian generating
functions on $\mathcal{M}$, and skew-symmetric Dirac generating operators
$D\in\mathcal{D}(M,S)$. The operator $D^2$ gives a new invariant of
$(E,\mathrm{g},\rho,[\cdot,\cdot])$, which generalizes the square norm of the
Cartan $3$-form of a quadratic Lie algebra. We study in detail the particular
case of $E$ being the double of a Lie bialgebroid $(A,A^*)$.
