Adjusting viewing angle without distortion.

I'm trying to adjust the viewing angle, without causing too much distortion. Playing with the camera viewing angle, and some code I found on here has the same effect. The code listed here manages to still distort the field of view. This post goes into modifying the projection matrix to achieve the effect, but doesn't list the formula needed.

This shows best what I'm trying to achieve:

pre-viewing angle mod:

Example of my scene:

Does anyone know what math I'd be looking for to achieve the desired effect?