When Nina Edwards Anker was growing up, her family spent summers and weekends in a shingled cottage in Southampton, N.Y.

An expansion of a carriage house originally designed in 1890 by Stanford White , it belonged to a genre typical of the Hamptons. But the version that Ms. Anker — the 48-year-old founder of the Brooklyn design company Nea studio — designed for herself; her husband, Peder Anker , a history of science professor at New York University ; and their two sons is anything but common. Located on the same property as her childhood cottage (which her brother now occupies), Cocoon House is a LEED-certified example of sustainable design that strays from many of the local norms.