It's not actually flat:

Hicks noted that, in reality, the mirror does not look like a disco ball up close. There are tens of thousands of such calculations to produce a mirror that has a smooth, nonuniform curve.

I imagine "tens of thousands of such calculations" is newspaper code for integration.

If someone assigned me the task of making such a mirror, I would start by treating the mirror like a monitor displaying a 3D scene, and calculate the vectors necessary to project a 45 degree view onto the surface of the mirror (i.e. start with a virtual camera behind the mirror such that it has a 45 degree field of view). I would then adjust those vectors so that the image will look like such a projection when seen by a viewer a few feet up and to the right, with the mirror occupying ~15 degrees of the viewer's FOV.

In other words, everything I need to know I learned from video games.

If I were asked to make such a device, I would build a flexible mirror whose curve could be adjusted by an array of attached screws. I'd fix said mirror to a car, and adjust the screws until the desired curve was reached. I'd then measure the curve and duplicate it in a rigid finished product.

It's actually amazing that it took a physicist/mathematician to think of this thing. I doubt I would have come up with the idea. It goes to show how domain knowledge shapes your perception of the world.