There is some discussion on the nLab on seeing the free cocompletion $\mathbf{Psh}(\mathbf{A}) = [\mathbf{A}^{op}, \mathbf{Sets}]$, as a pseudomonad. The Yoneda embedding $よ \colon \mathbf{A} \to \mathbf{Psh}(\mathbf{A})$ gives the unit and that the multiplication would be the unique functor $\mu \colon \mathbf{Psh}(\mathbf{Psh}(\mathbf{A})) \to \mathbf{Psh}(\mathbf{A})$ such that $\mu \circ よ = \mathrm{id}$. However, this creates some size issues.

On the other hand, composition of profunctors $p \colon \mathbf{B} \to \mathbf{Psh}(\mathbf{A})$ and $q \colon \mathbf{C} \to \mathbf{Psh}(\mathbf{B})$ can be defined (following Remark 5.6 in This is the (co)end...) as $\mathsf{Lan}_よ(p) \circ q$, which looks like composition in a Kleisli category for the (pseudo)monad.

Can we get the bicategory of profunctors as a Kleisli construction over the free-completion pseudomonad? Can we deal somehow with the size issues? Is there any reference where this construction is discussed?