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?
The Euler-Lagrange equation has a set of solutions. In classical physics, we consider only solutions which are c-number valued (or Grassmann-number valued if you include "classical" fermion fields); to quantise, we use solutions whose value is a linear operator from a Hilbert space to a Hilbert space (usually the same one). The trick is to ensure they satisfy canonical commutation relations, CCRs, (or canonical anticommutation relations, CARs, for fields satisfying Fermi-Bose stats) analogous to canonical relations. Just as Poisson brackets satisfy $\{q_i,\,p_j\}=\delta_{ij},\,\{q_i,\,q_j\}=\{p_i,\,p_j\}=0$ in discrete classical mechanics, and the first of these generalises in classical field theory to $\{\phi(t,\,\mathbf{x}),\,\pi(t,\,\mathbf{y})\}=\delta(\mathbf{x}-\mathbf{y})$, for a QFT we take $[\hat{\phi}(t,\,\mathbf{x}),\,\hat{\pi}(t,\,\mathbf{y})]_\pm=\text{i}\hbar\delta(\mathbf{x}-\mathbf{y})$ where we use a commutator or anticommutator accordingly (and of course also the $\hat{\phi}$-$\hat{\phi}$ and $\hat{\pi}$-$\hat{\pi}$ (anti)commutators vanish).
Linear ELEs lead to an especially simple result. Suppose a real classical $\phi$ satisfies a linear ELE with solutions $\phi=\sum_\sigma(\phi_\sigma a_\sigma +\text{c.c.})$ with complex coefficients $a_\sigma$ and complex functions $\phi_\sigma$; then the quantised counterpart is the Hermitian field $\hat{\phi}=\sum_\sigma(\phi_\sigma \hat{a}_\sigma +\text{h.c.})$. (You can obtain a similar expression for $\hat{\pi}$ from this.) The spacetime-constant coefficients are promoted to operators, the $\hat{a}_\sigma$ being annihilation operators so $\hat{\phi}| 0\rangle=\sum_\sigma\phi_\sigma^\ast\hat{a}_\sigma^\dagger| 0\rangle$ is a linear combination of $1$-particle states, with $| 0\rangle$ the vacuum ket.
