This might not be the most thorough tutorial, but I wanted to post it just to give readers an idea of what I did. To make a normal reversible sun hat, I found a tutorial on another blog. (Sorry, I searched for awhile and I cannot remember which one it was!) I picked Sesame Street fabric for one side and plain red fabric for the other side so I could add the Elmo face.
Before I completely sewed the entire hat together, I started working on the Elmo face so I could make sure it fit on the hat. I printed out a picture of his face that was resized to fit the hat.

I used some little scraps of fabric for the eyes, nose and mouth and two black buttons for the eyes.
Using my printout as a guide, I cut out the face shapes and ironed some fusible interfacing on the back to make them a little more stiff. I sewed the buttons onto the eyes by hand and then used the machine to do the rest.
I carefully pinned the face pieces onto the hat and then used a zig-zag stitch to embroider them onto the hat. Unfortunately, my sewing machine had a mind of its own that day, so the stitches don't look as nice as I wanted them to.
And the Sesame Street print on the other...
I love how it turned out and it's great to have two different hats in one!