Overlap map and frame in QGIS print composer

There is a way to do so, not very elegant, but useful.

In your print composer, add another map and choose the same scale as your main map. Move the second map to fit it in the correct position:

Then, put it over the main map in the "items" panel to see it over everything:

Here you can see the final result (I've done it in a couple of minutes just to show you, so it is not polished). It is a little tricky to put everything correctly. But once you get used to it, it is quite useful and easy to handle.

As proposed by Ed Rollason, I add my comment to the answer:

If you don't want your "overlapped map" to be rectangular, and wants to style it something similar to your example (where the extra space out of the border has the same shape as the continent), you can use svg shapes, as seen here:

https://smathermather.com/2013/04/09/quantum-gis-compositing-continued/

I hope it helps.