Turning a circle into a wrapped ring shape (not masking!)

One thing about the method I posted earlier is the sharp boundary between the stretched and round portions. Plus, the stretched portions look, well, stretched. If you wanted the image to "flow" around the straight sides, it would likely require planning how to cut it up properly, to modify and reassemble, as c.pfaffenbichler said.

When I first saw your Example pic, I thought it looked like the sort of thing Illustrator could do in an instant with its Art Brushes. Adobe had "Raster Art Brushes", if you will, in Adobe Sketch, but Sketch won't work anymore. As a mobile app, it relied on a touch interface; you would trace with your finger, and the brush graphic would fill the stroke. I still have several of these brushes in my Creative Cloud library they are orphans.

I think these would be great in Photoshop for stroking Paths, but they would probably be awkward to use with the Brush and a pointing device (mouse/tablet)

After making the ring via Polar Coordinates, set a horizontal guide on the center of the ring. Use the Single Row Marquee tool snapped to the guide, and Layer > New > Layer Via Cut. Select one of the (now separate) halves of the ring, drag it away from the other, and Layer > New > Layer Via Cut. Free Transform the Single Row layer to fit the space in between.

I think Puppet Warp or Illustrator’s Envelope Distort might provide options for this. 

Another option might be splitting up the panorama and threat the two curves as two halves for Polar Coordinates, but that seems pretty convoluted.