I know we all know what you're talking about but presumeably you want $\mathbb{R}$ as your codomain rather than the manifold.

Absolutely. This is an important one for me as well. I blame Tao's clarity of exposition for turning me into a dirty analyst. It is hard work to breathe the life of geometric and physical intuition into the skeletal technical parts of harmonic analysis and PDE, but he seems so often to do just that and I find it quite inspiring, particularly since analysis is often taught so listlessly.

I'm not sure I would have wanted to actually go to a course on such a topic. I did however read the Princton Companion in my spare time and would certainly recommend that interested undergraduates did so in order to put things in context. After all, an interested undergraduate should want to put things in context. However, in my opinion this is definitely the sort of thing that should be done in one's spare time. I think it's somehow too imprecise and intangible to be given as a course.