Ok does not sound so complex.
Since there is no big calender widget in the AUI library we took our time and found a good one which has a MIT license and sounds stable and well documented. The FullCalendar library seems to be the best choice.
We know now what we want to do - now let's see how we plan our components.
Kitchen Duty Overview Page Webwork Action - A page that can be viewed by all logged in JIRA users.
Kitchen Duty Overview Page JS Controller - Some JavaScript logic to display the calender with data from REST Endpoints on the page.
Kitchen Duty Overview Page REST Resource - An Endpoint that returns all weeks with users for a selected month.
Kitchen Duty Overview Page WebItem - A navigation link to Overview Page in JIRA top navigation.