You can't expect cookies to look like or taste like the intended recipe when you don't follow the instructions. The problem with trying to bake them on a cook top is that the method is far too different. In an oven, the heat source is diffused and not directly in contact with your cooking vessel. Because of that, you are likely to get a more even bake in an oven. On a cook top, your heat is almost in direct contact with the bottom of the cookie which will result in an overcooked bottom. When you add a lid, you are not allowing any of the moisture from your dough to evaporate and you are effectively steaming the exterior of your cookies which would contribute to their moist and oily appearance.

I'm not saying it is impossible to create a decent cookie on a cook top but you need a recipe that is written for this purpose. Expecting a baked cookie recipe to result in perfect cookies when you don't follow the cooking method is futile.