Going from classical field to quantum field operator?

Let us say I have a classical field theory, with a field $\phi(\vec r,t)$ which satisfies the relevant Euler-Lagrange equation for the Lagrangian $\mathscr{L}$. Is the general procedure (i.e. one that will work for any $\mathscr{L}$) for going from this classical field to the quantum field operators in QFT?

I know that one can look at commutators, but is this really enough to uniquely specify the field operators?