It will work on a shaded pole motor, but that may not be the best approach to take. The "controller code" you are looking for is something that companies spend 10's of thousands of $ to develop; I seriously doubt you will find someone to let you "borrow" it for free over the internet. Then you will need to integrate it into a controller, connect that to a power module, add protection, filtering, interface, packaging hardware etc. etc. Plus if you are going to buy less than 50,000 of them, they will be expensive to build.
That said, it would still be far less expensive for you to buy something already made. For instance, many many low cost (under $100) OEM VFDs are available now that will convert 1 phase 120VAC to 3 phase 240VAC variable frequency from 1/16 to 1HP. All you would need to do is change to a 3 phase motor. Easier to reverse too if that is a desired feature in your mixer.
If you insist on sticking to a single phase shaded pole motor, a company called Anacon Systems already makes a PWM drive for them.
"Venditori de oleum-vipera non vigere excordis populi"