How would I screenprint on a winter hat??

you will need a hat jig or press to print these properly. If you have the right equipment you print exactly like a shirt.

here is a great resource page full of industry links.
www.screenprinters.net/links.php

I would probably suggest embroidering the hat, depending on the design.

A friend of mine screenprinted a beanie and it looked like total sh*t. I assumed that said more about screenprinting knitwear than it did about his printing abilities. Would that not be the case then?

(can't say I've any intention of trying it anyway - personally I think it's one of those areas where embroidery is superior to screenprinting, but it's good to know your options just in case)

Thank you very mch guys!

Easy to do with a Hat Press and Plastisol Transfers with GLUE and extra thick patch like Ink spread. Perfect everytime.

As long as the Hat fabric properties do not change with Heat. It works fine.