@carp
Of the 38 courses I took as an undergrad, 31 directly applied to my major (EE) or my minor (math). That's 82%.
Even so, EE is a big field, so I took courses in Electronics, Communications, Microwaves, Power, Devices, and Computers Hardware.
All of my Master's courses have been focused on Simulation and Devices. I've taken 4 courses on Simulation and Math, 1 course on Device Physics, and 2 courses in Fabrication. So, you can see, even the Master's is very broad.
I started off thinking I wanted to do serious Simulating, but now I want to move more towards devices physics and fabrication.
So, I'd say I've narrowed down my field of interest as quickly as possible, but it's a big area (as I know ChemE is, as well), so it takes some time.