I have had a similar issue before in the spring when I used to take the winter off. I was not used to doing cardio the cold air and it would cause mucus to build up in my lungs and throat and give me a bad cough for several hrs. For me this would always go away with practice. If you ride in the cold you will get used to and not have any issues but it takes some time. For now you can use a mask to cover your mouth and nose which should alleviate the issue. Otherwise you will just have to push through but be careful because if your immune system is weak it could lead to pneumonia if not keep in check.

FYI,
I now commute and train all year round without any breathing issues including some single digit and negative temperature mornings. Tomorrow morning is supposed to be +4F with a wind chill of -4F. I don't expect any issues.