The book Boulevard of Broken Symmetries by Adriaan Schakel gives an excellent, if not exceedingly brief, overview of the path integral approach to perturbation theory. In particular, pages 47-58 give an excellent diagrammatic description of the partition function $Z[J]$, the generator of connected correlation functions $W[J]$, and the effective action $\Gamma[\varphi]$ ($\varphi$ is instead $\phi_c$ in Schakel's text). While I've found his exposition incredibly insightful, it would be nice if I could check my understanding against another textbook with a bit more detail. Unfortunately, most of the standard references I've checked (Peskin/Schroeder, Schwartz, Zinn-Justin, and a small handful of condensed matter field theory books) do not seem to have the same diagrammatic representation of these generating functionals as sums over all possible diagrams. Is there another reference which goes into these details?
I'm particularly interested to see if the Legendre transform $W[J] \rightarrow \Gamma[\varphi]$ can be understood diagrammatically. I think Schakel's book almost gets there: by writing each fully connected diagram in $W$ as a 1PI diagram plus propagators, and using the fact that the propagator is related to $\delta \varphi / \delta J$, I think it should be possible to trade external $J$'s for external $\varphi$'s. But I don't quite understand the full picture yet, and I especially don't yet understand how $\int J \varphi$ in the Legendre transform enters into the diagrammatics.
Apologies in advance if this question has been asked before, I would greatly appreciate any references!