My original question was to how to get method call counts. From the answers here, I gather that what I should be asking is what profiler output allows you to extract its information to get method call counts, and use that information in my integration tests.
Update: This is the application, I don't see how, or why I would want to add extra code to it for the sake of testing it. I don't mind writing code external to the site. We have integration tests that do not use the UI. Those tests pass, yet when accessed from the UI something somewhere has made an inappropriate call, circumventing caching. It's not noticeable on visual inspection. And up to this point, has never been tested in any of our UI tests (we use Selinium). I would imagine some sort of profiler would work, but I am not familiar with profilers. Much less how to log the information they produce and validate it within an automated test.