A CPT with only 1 extra field which is DATE
A View displays a calendar (monthly view) where we can navigate threw the months and years, and on the specific day, show the events assigned to that day.
Is there any documentation that you are following?
No
Is there a similar example that we can see?
No
This is not an events plugin, just a simple calendar to show posts created in a calendar view.
Is this easily done with Toolset?
This is an oft-requested feature, and we plan to work on adding it, but I can't say when. I know my colleagues have already started investigating 3rd-party JavaScript libraries for the calendar display, but haven't made any decision yet.
If you can't wait, then you would need to implement it yourself.
Google "javascript calendar library" and you will find there exist quite a few examples.
You would need to study the documentation to see what is required.
Generally, you would need to enqueue specific assets that the library requires (i.e. JavaScript and CSS files), and then you would need to ensure that the markup created by the output of a View to list your posts corresponds to that which is required by the calendar library.