Then I found double knitting.
Basically, you work two pieces at once, and the stitches alternate between the front piece and the back piece. You cast on and bind off with both colors held together. If you're working flat, one stitch on each edge is also worked with both strands together. You carry both strands together to the front or back of the work, but only choose one to loop around the needle. The result is double-sided--the same pattern on each side with the colors inverted.
I knit looser with this technique than usual--just something to watch out for.
I searched for some designs, and came across this one from The Purl Bee. With a table in Microsoft Word, I changed the pattern a bit to incorporate some birds (my sister's favorite animal). Add some ear flaps and a pompom and it's done! I am very pleased with how the hat turned out. My sister thinks it's warm enough, too!
(Sadly, the addition of the pompom means my sister can't wear this the other way.)
