Azure App Service: How can I determine which process is consuming high CPU?
May be you should go to webApp scm?
%yourAppName%.scm.azurewebsites.com;
There is a page, that can show you all process, that runned now on your web app. (something like Console > Process).
Also you can go to support page (from scm right corner).You can find some more info about your performance there, and make memory dump (not for this problem, but it useful for performance issues).
According to your description, I assumed that you could leverage the Crash Diagnoser extension to capture dump files from your Web Apps and WebJobs when the CPUs usage percentage is higher than the specific threshold to isolate this issue. For more details, you could refer to this official blog.