In my (admittedly limited) experience, cluster-to-cluster variation would make it very difficult to write tutorial with enough detail to actually run a job -AND- have it be useful outside of a specific university’s cluster. The main hurdles for me center on setting up the R environment (incl. where and how to install packages like brms or cmdstanr) and properly requesting resources via the job scheduler (varies depending on which your cluster uses: LSF, Slurm, PBS, etc).
Many university clusters have support staff who can be immensely helpful for working through any idiosyncrasies of setting up your R/Stan environment. There are good tutorials and example scripts out there for submitting an R script for many of the common job schedulers. For example, my university uses LSF and has prepared a set of publicly available written and video tutorials for running (parallel) R scripts geared towards beginners. To get this going with brms, you’ll mainly need to ensure that you’re matching the number of cores and threads to what you requested via the job scheduler.
For any remaining idiosyncrasies, it’s helpful to be ready to explain a general overview of what should happen to the support staff (i.e., brms makes code and data in R, sends them to Stan for translation, compilation with C++, and sampling). A cluster I used a few years ago kept failing when brms tried to compile the model. Explaining helped pinpoint the problem. For that particular cluster, the C++ compiler was deliberately not available on the compute nodes, meaning that I had to figure out how to go about Separating compilation & sampling with brms on cluster (kudos to the helpful folks here on the forum).
I hope I’m wrong and that others do know of good tutorials. If I’m being overly pessimistic, I wonder if folks with expertise would be interested in preparing tutorials to host on the Stan website tutorials. I’d love to see the same couple of models run with brms, cmdstan, and a Python interface crossed with the most common job schedulers.