What I came up with was...the parabola. So by using the properties of parabolas, we're going to cook a hot dog. Essentially, we're using math to cook hot dogs :D
I believe I've just proven that I'm a nerd? Oh well, I'm in good company here.
Step 1: How it works
The way it applies to us, is that the light that hits the parabola, will reflect back to one intersection point. That intersection point is called the point of focus. By placing the hot dog in the point of focus, all of the sun's rays that hit anywhere in the cooker, will reflect onto the hot dog...thus cooking it.
Click here to see how the light reflects
By the way...the shape of the entire cooker is a parabolic trough.