Does it get hard to get enough air, or do your lungs open up? Does it happen if you're doing those really hard efforts regularly, or only if you haven't done one in a week or two?

My asthma is very mild but strictly allergic, not exercise-induced. What happens to me is that if I've only been doing lower-intensity efforts, then when I go to my limit (like a long steep hill, a hard interval session or a 5K, say) I'll have symptoms that sound like yours (except my cough will be minimally productive, and it tastes really "stale"). My speculation is that the far reaches of my lungs get plugged with mucus, and only a really hard effort opens them up.

+1 on seeing a doctor.