Hi Kevin, even if the question is answered I would like to add a few remarks.
(0) the claim that
this quasi-isomorphism $U$ is compatible with the dg algebra structures on $T$ and $D$
is not exactly true. It is compatible only on tangent cohomology.
(1) I agree with Daniel and James that there exists a $G_\infty$-quasi-isomorphism between $T$ and $D$ (this implies compatibility at the level of tangent cohomology, and is strictly stronger). But until recently this was not known if Kontsevich's $L_\infty$-quasi-isomorphism can be upgraded to a $G_\infty$-one. A recent paper of Willwacher solves (in a positive way) this question (EDIT: Willwacher makes the comparison with Tamarkin's $G_\infty$-quasi-isomorphism in Section 10).
(2) the proof for smooth affine varieties is essentially the same as the one for smooth differentiable manifolds. Both rely
either on the exitence of a connection in the tangent bundle (see the papers of Dolgushev, e.g. this one).
or (equivalently) on acyclicity of sheaves of sections of bundles (see e.g. my paper with Michel Van den Bergh).
(3) references are
my paper with Dolgushev and Halbout for complex manifolds.
the above paper with Van den Bergh for a uniform approach to smooth, complex and algebraic settings (using Lie algebroids).
For a general smooth variety, though, instead of taking the Hochschild cochain complex of $A=\Gamma(X;\mathcal O_X)$, presumably we should take the Hochschild cochain complex of the (dg?) derived category of $X$. Is this correct?
One could do this, but one instead works on the sheaf level. Consider $T$ and $D$ as sheaves and prove that they are $L_\infty$- (or $G_\infty$-)quasi-isomorphic as sheaves of DG Lie (or $G_\infty$)-algebras.
(3+$\epsilon$) "Deformation quantization of Poisson manifolds, II" does not exist, but there is "Deformation quantization of algebraic varieties" (quite sketchy). You might also be interested by the very inspiring paper "Operads and motives in deformation quantization".