I am having the same issue, and a Google search lead me to this forum.
The original part is a HUA Li Lai "11 pins" "7 positions" using dual FZ31-9E parts with one of the 12 pins moved and probably a restrictor to reduce from 8 positions to 7.
I found this part on ebay for $24.99 USD
https://www.ebay.com/itm/144096921567?hash=item218cd887df:g:3YoAAOSwVBBg3cgV
The ebay part is a 8 position 12 pin instead of a 7 position 11 pin, which is the one needed for the Cuisinart TOA-60. I just ordered the eBay one. I didn't find the one that is acutaly needed for this toaster oven, but this will hopefully work if I hook it up the same way.
I pulled apart the original switch and it is totally destroyed internally. When I opened it, a metal piece fell out and everything was charred. I think the problem is that I switched the 7-position dial while the timer was running. That causes electrical arcing inside the dial and destroys it. In order to make this dial last, I will set it to the bake, air fry, or other settings BEFORE I turn the timer dial and make sure not to touch it unless I turn the timer back to the OFF position.
Hope this helps others.
