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Hhhm, that is very interesting! I'm not sure if it's a bug or not. I'll check out Launchpad in a few minutes.
I'm not sure exactly what steps you've taken, or in what order, as you've drawn this. I can stop the problem from occurring by selecting the hair object and clicking the Even-Odd Fill Rule button in Fill and Stroke dialog (roughly heart-shaped with a hole in it, top right corner of Fill tab). Then no matter which node you start to draw, the face doesn't get covered up.
What I can't figure out, is how the hair object exists as a compound path with 2 subpaths (generally concentric circles) and the center isn't filled, while the fill rule is still Non-Zero. And if you draw from the lower left node, the center remains not filled. But if you draw from the upper node, the center gets filled (as I would expect either way).
It might have something to do with the direction of the path. Did you perhaps use the Paint Bucket tool to create that path originally?
In my experiments, I can draw a doughnut with 2 concentric circles, and use the Paint Bucket tool to fill it. It is colored like a doughnut with the center not filled, but the fill rule is still non-zero. That's not what I would expect, because it creates 2 subpaths of a single compound path. If I draw the 2 paths and combine into a compound path, I have to change the fill rule to Even-Odd, to have the center not filled.
Interesting! I'll see if I can find anything at Launchpad

Edit
No issues reported in Launchpad, that I could find. Perhaps this should be reported? Well, I mean the issue with the Paint Bucket creating 2 subpaths of a single compound path, yet having still the non-zero fill rule.
I still don't know if the Paint Bucket tool created the subpaths in the op's case. I'll try some different search terms at Launchpad, and report what I find

Edit #2
No luck searching. Hopefully ~suv can help? Or anyone else?