Spherical Jigsaw Puzzle

I was thinking about what a spherical jigsaw puzzle would be like (e.g. like the wikipedia logo); in fact, someone here on thingiverse already made one; but one where the pieces form a thin shell. I was thinking about what happens if you make the pieces thick. If you do, you cannot just cut the sphere along the edges of each piece with a virtual knife pointing at the center of the sphere, because you wouldn't be able to get the pieces in and out.
So then I realized that if you cut the pieces