Looks like it just runs the LLM in a loop until it spits out something that type checks, prompting with the error message.
This is a cute idea and it looks like it should work, but I could see this getting expensive with larger models and input prompts. Probably not a fix for all scenarios.
I'm not familiar with how TypeChat works, but Guidance [1] is another similar project that can actually integrate into the token sampling to enforce formats.