The courses we took had labs starting from soldering physical components on boards (actually wire wrapping) to hardware design tools.
After graduating, I went to work for Mentor Graphics, (now a division of Siemens). They create chip-level design tools of all kinds, the big moneymaker back then was Calibre Parasitic Extraction.
They also sell PADS, a PCB design tool that was popular. I did lead gen for both these products and steered away from design and into software.
However, you could go to work for any of the electronic design automation (EDA) companies and ultimately learn the design tools at great detail which might be a way to jump into a position on a design team.
At the very least you would be in a position to build a professional network with chip designers.
Graduate degrees in physics and EE focused on semiconductor physics and manufacturing are also useful. I've known a few post grad/doctoral folks from nuclear engineering and optics that have careers in the field.