It's worth mentioning that Feynman's dissertation was on a Lagrangian formulation of quantum mechanics. However, just because Feynman thought it was interesting doesn't mean it's a good idea for the rest of us (although it's a relatively short 69 pages and an interesting read).

http://files.untiredwithloving.org/thesis.pdf

The path integral method of QED does make the Lagrangian for field theories easier.

https://en.wikipedia.org/wiki/Path_integral_formulation