One great way to deal with the challenge of grilling in extremely cold weather is to turn the temperature deficit into an asset.
Rather than having one lone person hover around a closed grill while others wait inside, cook on an open fire pit and let everyone gather around. The chef and the guests can savor the warmth of the fire, the aromas of the cooking food and the company of others.
People with the space and inclination can build a permanent fire pit into the ground. There also is a variety of durable but less permanent structures available.
Chimeneas, with a pot-belly fire pit and a chimney, are designed for cooking as well as warming. They are made of ceramic or of steel, copper or cast iron.
Other popular designs include fire bowls, rings and pits made of various metals. Some have screens; others are more open. Some are quite large. You can cook hot dogs or marshmallows on a stick, like on an old-fashioned campfire. Many models come with grill inserts for more elaborate cooking.
If you're sticking with a more conventional grill, there are ways to adjust to bad weather.