I think rotifer has the answer here . I also get this winter time and my problem is seasonal bouts of bronchitus.
You might try breathing in through the nose only to warm the incoming air. This would help a lot .
Also you could wear a mask or necktie scarf which would do the same job. The fact that this only happens on days when the air is cold is the clue to the problem
Try and get to see a doctor who specialises in sports and fitness problems you will get taken more seriously. 99% of doctors have no idea what serios cycling is. There idea of biking is a couple of miles down the road to the nearest pub and back
As an after thought if you are hammering it and have a few stiff hills on route then that could induce your symptoms if you are pushing quitwe hard. If you see the Tour de France riders giving interviews straight after a race they always cough. Exertion of that magnitude actually damages the tiny aerololes of the lungs and causes fluid build up and some slight bleeding from broken blood cappilaries. This would be more pronounced if you were suffering form a chronic lung infection.
I'm sure we have some fitness experts here that can explain this better than I