EDIT: So, I was able to get this to work... although, it slightly more difficult than just specifying a path..
You can accomplish this by specifying a volume in docker-compose.yml.. The path to the directory (on the host) is labeled as device in the compose file.. It appears that the root of the path has to be an actual volume (possibly a share would work) but the 'destination' of the path can be a directory on the specified volume..
I created a new volume called docker on my machine but I suppose you could do this with your existing disk/volume..
I am on a Mac and this docker-compose.yml file worked for me:
version: '3.7'
services:
nodemon-test:
container_name: my-nodemon-test
image: oze4/nodemon-docker-test
ports:
- "1337:1337"
volumes:
- docker_test_app:/app # see comment below on which name to use here
volumes:
docker_test_app: # use this name under `volumes:` for the service
name: docker_test_app
driver: local
driver_opts:
o: bind
type: none
device: /Volumes/docker/docker_test_app
The container specified exists in my DockerHub.. this is the source code for it, just in case you are worried about anything malicious. I created it like two weeks ago to help someone else on StackOverflow.
Shows files from the container on my machine (the host)..

You can read more about Docker Volume configs here if you would like.
ORIGINAL ANSWER:
It looks like you are trying to share the build directory with your host machine.. After some testing, it appears Docker will overwrite the specified path on the container with the contents of the path on the host.
If you run docker logs my_laravel you should see an error about missing files at /var/www/site.. So, even though the build is successful - once Docker mounts the directory from your machine (./site) onto the container (/var/www/site) it overwrites the path within the container (/var/www/site) with the contents of the path on your host (./site) - which is empty.
To test and make sure the contents of /var/www/site are in fact being overwritten, you can run docker exec -it /bin/bash (you may need to replace /bin/bash with /bash).. This will give you command line access inside of the container. From there you can do ls -a /var/www/site..
Furthermore, you can also pre-stage ./site to have a random test file in it (test.txt or whatever), then docker-compose up -d, then run the same commands from the step above docker exec -it ... and see if the staged test.txt file is now inside the container - this gives you definitive evidence that when you run volumes, the data on your host overwrites data in the container.
With that being said, doing something like this and sharing a log directory will work... the volume path specified on the container is still overwritten, the difference is the container is writing to that path.. it doesn't rely on it for config files/app files.
Hope this helps.