The assumption that aspheric lenses create "abnormal" projections is incorrect.
There is nothing "normal" in normal spherical lenses, except that we can produce them cheaply. You don't get rectilinear projection just because you used spherical lenses, you have to struggle for accurate rectilinear projection (if that's your goal). Pinhole is automatically rectilinear, that's simply geometry, but once we depart from thin lenses, there are many mutually influencing and nonlinear parameters and many degrees of freedom. Change lens curvature, spacing, index of refraction - and you get different aberrations, projections, distortions, everything. Apsherical lenses is only one step further, not something fundamentally different.
By the way, this is Nikon US Patent 3,737,214 - fisheye lens consisting of only spherical lenses.
How can you design such thing? The basic ideas come from great predecessors and then, nowadays you use computer application that calculates everything and can automatically optimize certain parameters following the given criteria.
For example this is from OSLO (commercial, free trial available):
Playing with such software is great way to feel how the lens design really works (even if you only use it for "breaking" existing examples and watching how the performance deteriorates)