Thanks so much for your response! The references are exactly the kind of thing I was looking for.

Do you know where one could find a precise statement of this fact? I'm familiar with Block's result embedding $D^b(X)$ into the derived category of the Dolbeault dga of $X$, but the notion of a CY algebra predates his result.

I get that CY is an additional structure on a dga. What I am seeking is some historical/geometric reason that noncommutative CY's are modeled on dgas as opposed to ordinary algebras. In the commutative world, the coordinate ring of a CY variety need not be a dga (I think?), and much noncommutative geometry is done with ordinary algebras.