eumorphic projection

Dear all, 

I'm struggling with the following and I hope someone here is interested to think along.

My aim is simple. To create a map projection of an egg. The projection i am particularly interested in is the Boggs Eumorphic Projection.

So far a have scanned a lapwing egg with Agisoft PhotoScan and managed to make a 3D model of it.

Something went wrong on the right side which I will fix later. I had to start testing things and I only had a broken egg.

My aim is not to make an Unroll Surface (which is not too difficult) but much more something like a map like this. 

My final 'product' will be a 2D print; a 'world map' of an egg.

Grasshopper is new to me. Also Blender is new but there I saw some tutorials where they show quite a strong UnWrapping tool that could help.

But maybe one of you have a much simpler idea how to approach this. 

Hope to hear from you.

Best, Rutger