The company, which has a factory in China and another in New Jersey, has so far manufactured two prototypes: the one on display at Mecox Gardens, which will be delivered to Ms. Burn and her husband, Spencer Grimes, a financier, after the summer-long open house ends; and the Cocoon Cabin, the same size but with a different floor plan, at Mr. Burch’s Southampton home, where it serves as his pool house and guest cottage.

Cocoon9 is selling three models, all built as single modules and designed so as to be easily transportable by flatbed truck. Both the Cocoon Cabin and the Cocoon Studio measure 12 by 40 feet and are similarly priced. The Cocoon Lite 20, at 8 by 20 feet, costs between $75,000 and $95,000. Made of steel that can withstand winds of up to 150 miles an hour, they are insulated so as to require minimal air-conditioning or heating. The houses also have several environmental features, such as LED lighting and nontoxic materials.

Despite their contemporary, geometric shape, these buildings suit even the traditional architecture of the Hamptons, “where pitched roofs and cedar siding are the norm,” as a blog post on the Mecox Gardens website noted. The exterior of the Cocoon Studio is a wood-plastic composite, while other models have exteriors of carbon stranded bamboo siding. Cabinetry can come in several finishes, including bamboo or a matte or glossy white lacquer, while flooring is either cerused oak or vinyl with an oak grain. The fixtures can also be tailored to taste, from Venetian bronze to brushed stainless steel, with bathrooms that feature marble mosaic or glass tiles.

The houses, which take four months to manufacture, also feature distinctive touches like 360-degree TVs that can swivel from one room to the next and showers made of “smart glass” that changes from clear to opaque when in use. Buyers must build their own foundations on which to attach the buildings, secure the proper permits and connect the utilities themselves, the costs for which can vary, but may add up to around $30,000, according to the company. A crane is also needed, which can run an additional $10,000, while shipping costs can range from $5,000 to as much as $40,000, depending on the state. Once the home arrives, it can be hooked up in a matter of days.

For Ms. Burn, the style of the Cocoon Studio at Mecox Gardens matched her house, designed by the architect John Black Lee, seamlessly. “Our house has such a clean look and we wanted to honor that architectural aesthetic,” she said.

So far, Ms. Burn is Cocoon9’s first and only client, but after the soft launch at Mecox Gardens, interest is growing, according to the company. The hope is that these buildings will not just adorn the backyards of expensive homes, but that hotel developers will buy several buildings and install them along a beach in the Caribbean, for example, to create a near-instant eco-resort.