Goldenseal,
Great Pyrenees, Anatolian and Maremma sound like good breeds for you to consider. With the deep cold you have in your area I would look at the GP and the Maremma first, due to their heavy coats.
I have GPs to protect my place. Lost chickens, cats and my wire fox terrier to coyotes in the past, now with pyrs patroling, I haven't had a 'yote on my place since.
Got my pyrs from a family farm that raises goats and has pyrs to protect their herds. They only breed their dogs when they need more or when friends need a pup or two. In other words they are NOT breeders, wanting big bucks for show dogs. Their dogs have no papers, don't cost an arm & leg.........but their dogs do what the breed was intended to do, that is the dogs haven't had the instinct bred out of them. (GP have a wonderful history that goes back over 500 years, protecting shepherds, their flocks as well as royalty.)
I would also encourage you to consider getting at least 2 dogs. They really do work better as a team, and when dealing with a pack of coyotes, they can back each other up. I've heard of folks, who only had one dog, losing it to a 'yote pack.
Got my pyrs as puppies (different ages) and taught them who to look after, that is, who was family.....be they 2 legged or 4 legged. The older dogs help to train the puppies. These 'gentle giants' can be excellent family dogs, trusted around anything YOU consider family.........but he11 on wheels for anything/one that doesn't belong.
Best of luck in your search for a LGD.