Wet and Wash
To start cleaning your hat, you'll moisten it in clean, cold water. Fill a large bowl with cool water to quickly submerge your hat. If you like, you can buy a cleaner that the UGG boots on sale recommends. Otherwise, look for a cleaner designed to gently clean and condition leather and natural fabrics such as sheepskin. Use a damp sponge to gently apply the cleaner, but don't scrub too hard. Getting too aggressive with your sponge can make the fibers break down too quickly.
When you have your Ugg hat as clean as it will get, it's time to rinse. You'll use cold water again for rinsing. Continue to rinse until the water coming off your fabric appears clear, not cloudy. If you still see some dirty areas, go back and scrub again with cleanser. Resist the urge to crumple and wring out the extra water -- instead, gently press a towel against the fabric to remove the excess moisture.
You definitely don't want your hat to lose its shape. So place UGGS on sale over a towel and stuff it with a washcloth to imitate your head shape while it dries. Here's a warning: It will take a while for your hat to dry. But that's to be expected since the fabric is so thick and keeps you toasty. When it finally dries, spray on a suede or nubuck protector so your hat will be waterproof.