Sorry I was unclear- the cooktop was on an island in the center of the kitchen, with no stove underneath. The stove is @2' away.
Even though it is my fault, we, as a family, are all contributorally negligent. We had gotten very lackadaisical about putting stuff on the cooktop. I bet if it had been one of those old coil type tops, it never would have had the microwave lid resting on it in the first place. But that black glassy top is soooooooooo inviting. Even though I didn't do it this time, I've left those lids on the cooktop myself probably a hundred times.
On the bright side, this thing should have a serious vent on it. The Kitchenaid that it is replacing had a very weak vent that couldn't even keep up with a hint of smoke. This meant if something was a fairly smoky dish, we'd often end up opening a window or some such. Even so, a downdrafting vent won't be nearly as good as an overhead one of comperable power.
Big lesson: if you are ever in a position to build or renovate a kitchen that features an island cooktop, SPEND THE EXTRA MONEY UP FRONT and get one that vents from above. With lighting. You will save yourself money and headaches in the long run.