This is a consequence of Theorem 3.3 in Gelfand and Gelfand's Tensor products of finite and infinite dimensional representations of semisimple Lie algebras; the projective cover of $M_{\lambda}$ is obtained by translating $M_\chi$ by an indecomposable projective functor.
Here's my "by hand" proof, which I'll leave here, since I wrote it out before bothering to look up the reference: The important point here is that $E\otimes M_\chi$ has a Verma filtration where $M_{\chi+\nu}$ appears with the multiplicity of the weight $\nu$ in $E$. The highest weight is at the bottom of the filtration, the lowest at the top.
Consider the case when $\lambda$ is dominant; this follows by the usual argument that translation functors give equivalences between blocks of category $\mathcal{O}$; Let $E$ be a f.d. representation with $\lambda-\chi$ extremal. Thus, $M_\lambda$ appears in the filtration on $E\otimes M_\chi$, and no other Vermas in the same block appear, so it's a summand and thus a quotient.
Thus, it suffices to replace $\chi$ with an arbitrary dominant weight; in particular, we can assume that $\chi-\lambda$ is dominant. Now choose $E$ to be the representation with lowest weight $\lambda-\chi$. Then $M_\lambda$ is the quotient of $E\otimes M_\chi$ by the submodule generated by all vectors of weight $>\lambda$. Thus we're done.