The shortest day of the year is around December 21, but that's not the coldest day.
The arctic region remains in complete darkness from late November until the second or third week in January.
Cold air continues to build across the Arctic region as it sits in near or total darkness.
From time to time blast of this Arctic air will spill south. But since (during this time period) there is no sunlight, there can be no warming.
Once the Arctic region starts to get a little sun each day, the cold can be gradually modified.
Each subsequent blast from the Arctic, with time, becomes less intense. Thus, average high and low temperatures in the winter tend to bottom out toward the end of the "darkness" period across the North Pole/Arctic.
This is why there is a lag between the shortest day and the coldest days of about a month.