How to put Embroidery on Straw / Synthetic Straw Hats

Sunny days are here, and so are big-brimmed floppy hats. There's no better way to keep the sun off your face and neck while outdoors. And of course, they must be embroidered.

But what kind of designs work best? What kind of stabilizer, should a topping be used, and how in the world does one hoop a big ol' brimmed hat? I bought several hats and tested more than a dozen different design and stabilizer combinations to find how to get top results.