My approach involves Deform > Twist - you can pick the original curve used to revolve the shape, and apply 2 different twist angles using the same centerline as the main object, then rotate one slightly to open one end. Then using a combination of project and trim you can extract two new profile curves to sweep along the two twisted/helix curves created earlier. This swept surface closely (but not exactly…) follows the surface of the dome shape. Next, you shell it (centerline, both-sided) and work from there. I cut off some parts using boolean diff with a straight line for example.

The technique of using Twist on a revolve profile works in most situations where you need a helix to follow such shapes exactly. I think it supersedes DannyT's technique in many cases now that Twist is available to us.

Turns out you can even use the twisted curve to directly Trim the surface, yielding an exact segment of the shape you can shell and work with as well. This is what I've done in the attached image.

I've attached an example below - if you watch the styles closely you should be able to figure it out. Let me know if you need some more detailed instructions!

/ Fabien