This is a surprisingly good question. "Good", because it is indeed true that there is this very general prescription for quantization; and "surprisingly" because, while the general idea has been around for ages, this has been understood in decent generality only last year!
Namely, on the one hand it is long appreciated in the context of quantum mechanics that what physicists sweepingly call "canonical quantization" is really this: the construction of the covariant phase space as a (pre-)symplectic manifold, and then the quantization of this by the prescription of either algebraic deformation quantization or geometric quantization.
In contrast, it has been understood only surprisingly more recently that established methods of perturbative quantization of field theories, especially in the guise of Epstein-Glaser's causal peruturbation theory (such as QED, QCD, and also perturbative quantum gravity, as in Scharf's textbooks) are indeed also examples of this general method.
For free fields (no interactions), this was first understood in
- J. Dito. "Star-product approach to quantum field theory: The free scalar field". Letters in Mathematical Physics, 20(2):125–134, 1990.
and then amplified in a long series of articles on locally covariant perturbative quantum field theory by Klaus Fredenhagen and collaborators, starting with
- M. Dütsch and K. Fredenhagen. "Perturbative algebraic field theory, and deformation quantization". In R. Longo (ed), "Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects", volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001.
Curiously, despite this insight, these authors continued to treat interacting quantum field theory by the comparatively ad hoc Bogoliubov formula, instead of similarly deriving it from a quantization of the (pre-)symplectic structure of the phase space of the interacting theory.
That last step, to show that the traditional construction of interacting peturbative quantum field theory via time-ordered products and Bogoliubov's formula also follows from the general prescription of deformation/geometric quantization of (pre-)symplectic phase space was made, unbelievably, only last year, in the highly recommendable thesis
Just read the introduction of this thesis, it is very much worthwhile.
(I learned about this article from Igor Khavkine and Alexander Schenkel, for which I am grateful.)
In a similar spirit a little later appeared
which disucsses the situation in a bit more generality than Collini does, but omitting the technical details of renormalization in this perspective.