Ha! I found some stuff out about them! They are called papakhas.
This is from some russian site:
Headgear symbolizes dignity
This traditionally applies to both sexes in Chechnya.
Each Chechen man treasures his 'papakha' sheepskin hat. "As long as you have a head, you must have a papakha on it"; "If you have nobody to keep counsel with, seek advice from your papakha hat", old proverbs say. The papakha is worn even indoors, only the 'bashlyk' hood is pulled off when the wearer comes in.
For important social occasions and for appearances in town, each Chechen man has a particularly smart papakha. Young men sport smart and stylish papakhas. Each man, young or old, takes utmost care to keep his papakhas neat and clean.
Knocking off the papakha from someone's head amounts to a horrible offence. Tampering with a papakha that is briefly abandoned by the owner may cost you friendship with him. Taking off the papakha and throwing it to the ground spells readiness to go any length in a dispute, quarrel or battle.
A Chechen woman can stop a duel by throwing in her headscarf. But a man's papakha is supposed to always stay on his head. A man who takes off his papakha in a sign of humility or request is despised as slave. Only a bid to end a vendetta forgives publicly taking off a papakha hat.
The world-famous Chechen dancer Makhmud Esambayev, Hero of Socialist Labour and People's Artist of the Soviet Union, was an archetypal Chechen 'kyonakh', or knight. He called his papakha 'my crown' and not even a meeting with a head of state could make him remove it from his head. His papakha made an unparalleled and very conspicuous presence on the floor of the Soviet legislature. On noticing it, the Soviet leader Leonid Brezhnev would murmur in content that "Makhmud is here, it's time to call the session open."
In 'My Dagestan', the great Dagestani poet Rasul Gamzatov referred to the papakha hat as a hallmark of the North Caucasus in all real and symbolic aspects: "The great dancer Makhmud Esambayev performs dances from all over the world but he never takes off his Chechen papakha from his noble head. Similarly, may the whole body of my diverse poetry always wear a North Caucasian papakha on the top."