Winter is caused by the tilt of the Earth's axis - 23.5 degrees relative to the plane of the ecliptic. That means the north pole is in daytime for six months and nighttime for another six. The entire polar cap has varying amounts of night and day with the Arctic Circle being the limit of locations experiencing any day of continuous daytime or nighttime. (The same is true for the south pole and the Antarctic Circle, with the seasons reversed.) Again, this is due to the tilt of the planet and will not change with any changing climate. In other words, there will always be a winter.
As a result, there is a huge mass of air that will be sitting in extended nights for months on end and will get cold. Even if it warms up (and the Arctic is warming faster than any other part of the planet), it will still be really cold - negative 40 is warmer than negative 50, but still cold. Given a stable system, that cold air mass would sit in the Arctic and we would never see it. But, the system isn't stable. Even without climate change, there are currents and systems that mix up the lower atmosphere (the troposphere) and move air around. Put extra energy into the system and this mixing becomes more pronounced. It takes a lot of energy to move air around.
And, that brings up the point - how much energy does it take to move an Arctic air mass? We can do a quick back-of-the-envelope calculation to see it is an extreme amount. Take a look at the image below. This is a plot of the temperature anomaly. You can see the cold air mass over the eastern US in blue. You can also see the hotter than normal air mass over the Arctic, Europe and northern Asia in red. When that cold air moved out, something had to move in to take it's place.
I'm going to estimate that blue area, the Arctic air mass, is about 3000 miles across. That's a radius of 1500 miles, about 8 million feet. The area of a circle is pi*r^2, giving us an area of about 2 x 10^18 square feet. There are 144 square inches per square foot, so this mass has an area of roughly 2.9 x 10^20 square inches. Standard atmospheric pressure is 14.7 pounds per square inch (you were wondering where I was going, weren't you?), meaning this cold air mass weighs about 4.2 x 10^21 pounds. That's a lot of weight.
Like I said, left to itself, it would just sit there. We need energy to move it. We're not going to consider all of the complications, such as friction with the ground, internal friction, changing pressure going over landforms, moving air in to replace it, etc. We're simple going to look at the kinetic energy of this mass when it moves. The equation for kinetic energy is one-half*m*v^2. To do this properly, let's change our weight to metric. There are 2.2 lbs per kilogram at the Earth's surface (we need that stipulation because we're going from weight to mass). Our calculated weight has a mass of about 1.9 x 10^21 kilograms. Let's use a standard velocity of 1 meter per second (2.2 mph). We can then multiply our result to get a value for different velocities.
Using those numbers, the kinetic energy of our air mass is about 10^21 joules. In comparison, a one megaton nuclear weapon has a yield of about 4 x 10^15 joules. That means this air mass would need the energy equivalent of around 250,000 one megaton nuclear weapons to move.
And, that is at a rate of only 2.2 mph (1 m/s).
Give the air mass a typical velocity of around 30 mph (13.6 m/s) and you need 1.3 x 10^22 joules of energy - about 3.4 million one megaton nuclear weapons. And, that's just the kinetic energy. Throw in all of those complications we didn't consider and you can the total energy involved is going to be much, much larger.
All of that energy MUST come from somewhere.
So, next time someone jokes about how we need more global warming on a cold day, you can point out how global warming makes it more likely we're going to see cold days like this.