Men in the Northeast tend to do one of two things concerning their scalps and ears on very cold winter days: nothing or barely something... similar to what they do when it rains.
Further north (Maine to the Dakotas and upward), men have a bit more sense concerning the cold and their heads. I have a theory that each person has "their temperature" on either end of the thermometer. These are the points below or above which all care about style and presentation abruptly end. For me, about -10 Fahrenheit trots out the "snow ninja/nuclear refugee" look, while my wife's thermostat engages at anything below 40. All of this aside, the question remains: How does a man dress in harsh winter, concerning his head?
It seems that in the US, we are very bad at wearing hats (this generation), so one should first look to the cold weather hats sanctioned by uniforms. The US Coast Guard Academy, the US Post Office, Police Departments, etc., all prefer the aviator-style hat with fur or wool in/suede or leather side out. Obviously, a relatively plain dark colored watch cap is always safe in cold, but if you have to be seen outside in these low temperatures (parades, dedication ceremonies, caroling, etc.) and you are required to dress, a better hat is called for. Fur.
If fur is not appealing, one can go synthetic if you dare, or try out lambs wool in one of the shapes ranging from envelope to stove-pipe.
The Russian "ushanka" (ear hat) is also a prized winter option, and I'm seeing more of them around various financial districts. Be sure that you remove all medals, badges, insignia, etc., or you'll look like a fool and will be the recipient of Red-baiting from the old-timers. President George H. W. Bush wore one during the painfully cold inauguration of President Obama, rocking a purple scarf and wool turtleneck... no tie.
Here, President Ford still looks patriotic wearing his.
The ear-flaps on the ushanka do not need to be deployed to be warm. If they fasten atop, simply loosen the ties and lower the fold to cover half of your ears, and it is remarkably warm. If it gets too cold, and you drop the flaps fully, you will be warm beyond belief, but won't hear a thing.
Less dress options include the hunting style hats such as my favorite wet/cold duck hunter ("Jones Style") from LL Bean:
What you don't see are the fleece ear-flaps/half-band that fold down for insane warmth. Drop the back brim down (if your collar is high) and it's fantastic in hard rain. It is warmly lined, and I cut off the tag you see in the picture. $40 or so, but with a lifetime warranty as usual per LL Bean. They also offer a gray wool one (dressier?) for under $30.
I'm not sure why men have such strained and dysfunctional relationships with hats, either adopting a single one as their "trademark", wearing them indoors, at a table, reversed, or just not wearing one when they are needed (sun, snow, etc.). I don't want this to sound like a retro-nostalgic plea for all men to start wearing hats again at all times, but the abstinence and hard-line approach to hat policy is childish.