Unfortunately my only exposure to this is a University course I took called Computer Intensive Methods in Mathematical Statistics[1].

It was a really good course, but at the time I was not in a mental space where I could get the most out of it, so I'm kind of hoping someone else can point to a solid textbook or something.

[1]: https://www.kth.se/student/kurser/kurs/SF2955?l=en

I liked the book "Statistical Mechanics: Algorithms and Computations" by Werner Krauth. It's oriented toward physics, but IIRC he links Monte Carlo simulations to the path integral formalism.

There is also a MOOC (same title, same author) on Coursera that I found interesting. It doesn't not go to the same depth as the book however. They are more complementing each other, the book for theory, the MOOC for implementation exercises.

this looks wonderful https://www.coursera.org/learn/statistical-mechanics

I don't know if I can plug-in here some of the materials I prepared some years ago. It might help to bridge/connect the use of Monte Carlo techniques and its relations with integration:

https://github.com/ChristopheRappold/MCandPython/blob/master...

Also just to mention something else that is very interesting related with MC : "quasi-Monte Carlo techniques" (small example in https://github.com/ChristopheRappold/MCandPython/blob/master... ). For people interested in MC, just take a look to those keywords.