That's the problem. What you're asking is mathematically impossible (the sphere <-> plane transformation always distorts some aspect of the elements, such as shape or size).
Having said that, it is possible to find a projection that results in the plane map that you're working from and back-projection your plane map onto a sphere. There will be distortion on the spherical version, but how much will depend on the source map. Ultimately, you will need to decide which is more important to you: an undistorted globe view or an undistorted plane view.
Here's a very quick example of taking your map, assuming that it's already in the orthographic projection (that is, it's on a globe being viewed from infinity) with the bounds being set at about Europe (your second map shows that I didn't quite get it right, but I didn't put a lot of work into getting it there). Then I took that reprojected image and plopped it on top of a map of the Earth in several projections to show you the sorts of distortions that are just unavoidable.
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