Cookies – How to solve under-baked cookie centers

My guess:

With a little oven you probably don't preheat as long, and when you open the door you let ALL the hot air out. You need to reheat the air in the oven after the cookies go in for them to cook. Because the hot walls and rack are much smaller (and probably not as well heated to start with), it's harder for them to reheat the inside of the oven.

So the temperature is lower when you start the cookies. Some (maybe a lot) of the cooking time is spent getting the oven, the baking sheet and the air back up to temperature. Cooking them longer or at a slightly hotter temperature should help with this.

Chris hinted at it -- to get the cracks, you need to have the top set early, so it's not going to expand while the middle's still trying to rise, and there's still enough leavening left to break through the crust that formed.

You have a few things that can go wrong:

1. The oven isn't hot enough. (it needs to set the top before the middle's fully risen)
2. Not enough leavening (it needs to be strong enough to crack the top once it's set)
3. Using a single-acting baking powder (double acting gives extra rise when it gets heated)
4. Not creaming the fat long enough (creaming cuts little air pockets into the fat, which means the leavening has to do less work)
5. Making too thin of cookies (you need enough mass in the middle so it finishes cooking after the top's set).
6. Mixing the dough too much (you don't want to develop gluten; it needs to be tender enough to crack easily)

As Jennifer mentioned -- old baking soda can be a problem. I'd also consider investing in an oven thermometer to make sure that it's really providing the temperature the dial claims, as those are the two things that are going to be the most difficult to judge if they're a problem or not.

You'll also want to make sure the oven is properly pre-heated (so it doesn't loose all its heat when you open the door to put the cookies in), and avoid opening the door again 'til it's time to pull them. If you have to rotate them (as the oven cooks unevenly), do it as quickly as possible so the oven's not open for very long.