What is the small peaked roof construction on the high side of a chimney called?

The construction feature referred to in the question is known as a cricket. A cricket is a small peaked roof structure designed to divert water away from the chimney, preventing it from pooling on the roof. It is typically installed on the high side of a chimney and helps channel water down the roof slope, mitigating the risk of leaks and water damage.

This design is particularly beneficial in areas with heavy rainfall or snow, as it promotes proper drainage. By effectively directing water away from the chimney, a cricket enhances the longevity of the roofing materials and the chimney itself. In contrast, features such as a pitched pocket, parapet, or dormer serve different architectural purposes and do not specifically address the issue of water diversion around a chimney.