My feeling is that this is really about set-theoretic complete intersections.
Let $X={\rm Spec}\\, A$ be a noetherian affine scheme such that every irreducible subscheme of $X$ is a set-theoretic complete intersection. In other words, for any prime $\mathfrak p\subset A$, there exist a set of elements $x_1,\dots,x_t\in \mathfrak p$ such that $t={\rm ht} (\mathfrak p)$ and the $x_1,\dots,x_t$ generate a $\mathfrak p$-primary ideal with $\mathfrak p= \sqrt{\( x_1,\dots, x_t\)}$, or equivalently the zero set $Z(x_1,\dots,x_t)=\overline{\{\mathfrak p\}}$.
In this case, take $M=A/\(x_1,\dots,x_t\)$ has the property that ${\rm Ass} (M)=\{\mathfrak p\}$.
If in addition $A$ is CM, then so is $M$ and then its projective dimension satisfies that $$ {\rm pd} (M)=\dim A-\dim M= {\rm ht} (\mathfrak p). $$
I suppose the next step is to look at an affine scheme with an irreducible subscheme that is not a set-theoretic complete intersection and see what happens there.