The key to avoiding this common problem is to use lightweight, thin underlayers made of merino wool or polyester, usually referred to as “thermo layers.” (Read: No cotton!)
These thermo layers help you stay warm or cool by regulating the moisture on your skin really well. If you are too warm, they conduct heat and sweat away from your body quickly to help you stay cool and dry. If you’re cooling down, you won’t get too cold because you’re not wet. This is what cross country skiers do to regulate body temperature.
Depending on how cold your winters get you could go for the sleeveless or long-sleeve undershirts. Also, the non-wool versions don’t keep you as warm as the wool so they can be used in a much wider temperature range. So it all really depends on how cold your winters get. A thermo-layer based cap underneath your helmet helps in freezing temperatures as well. Last but not least, when you have to deal with cold winds, rain or snow, applying lotion to your face or neck really helps reduce the sting dramatically. I have been really displeased with most lotions but have found Vaseline’s Moisture Control Lotion to feel really good. Hope that helps!