For many years, Matthew and Jennifer Miller had an idyllic Hamptons escape for summer weekends away from their sweltering apartment in Manhattan: a quaint, 400-square-foot studio above a detached garage at the Bridgehampton, N.Y., home owned by Mr. Miller’s mother.

“For us, it was perfect,” said Mr. Miller, 40, the founder of StudioLAB, a New York-based design firm. “We were just blessed to have summer time out in the Hamptons.”

But when they added two growing children to the mix — Sophie, now 7, and Miles, now 4 — the small apartment began feeling cramped, and the arrangement seemed a little less ideal. “We realized that if we were going to continue to come out, we needed our own space,” Mr. Miller said.

When they began hunting for a place of their own in 2018, their goal was to find a house for less than $1 million. But they were dismayed to discover that even homes in rough shape, with few redeeming features, were well beyond their budget.

“The market was so high and out of control,” said Ms. Miller, 40, a lawyer specializing in real estate at Cozen O’Connor. “There were a lot of old houses here that needed a lot of work.”

“Even a teardown was north of a million dollars,” added Mr. Miller, who, despite being a designer who doesn’t flinch at the idea of demolition, worried about how they could afford to both buy and build a new home.

Eventually, they found a house in Sagaponack that looked promising: a 1985 saltbox of about 1,400 square feet with a pool, on about 1.3 acres of forested land. The asking price was $1.1 million, but the seller’s agent told them there was already an offer on the home. Rather than engaging in a bidding war, the Millers saved the listing to their Zillow account and moved on.

Over the following weeks, as they received Zillow alerts indicating that the property was still on the market and the seller was gradually dropping the price, they sensed opportunity, and negotiated the price down to $900,000.

By the time they closed that fall, Mr. Miller had plans and permits in hand for a major renovation and addition, with the goal of having the house move-in ready by the following summer. “You’ve got to get into the ground before it gets cold,” he said of his rush to start construction, “because you can’t pour concrete if the ground is frozen.”

Mr. Miller’s plans included clearing much of the forest to create a lawn for his children to play on, and expanding the home’s size to 3,100 square feet. He did so by retaining most of the original house as one wing containing three bedrooms and a playroom; mirroring its shape with another wing containing the kitchen, dining room and master suite; and connecting the two spaces with a third volume in the middle — a double-height living room.

To make enough space, he designed a new pool for the freshly cleared backyard, and used a hole left by the old pool to hold the foundation for his addition. He also designed a simple, detached two-car garage.

In terms of style, “I gravitated toward this modern farmhouse,” Mr. Miller said. “Because I was working off the existing structure and roofline, I knew it was going to be a gabled house, not a flat-roof modern box.”

Still, he wanted to give it a contemporary edge: “I didn’t want it to look like a 1900s farmhouse.”

Outside, he kept the detailing clean and crisp, with a minimalist color palette: white cedar-shake siding punctuated by black windows and doors. Inside, the home has a bright, airy feeling, with whitewashed white-oak floors and plenty of white-painted shiplap paneling.

To keep costs down, Mr. Miller served as the general contractor and managed all the subcontractors himself. He also called in favors from suppliers, getting discounts on materials like the white-oak flooring and tile, as well as the plumbing fixtures and a concrete sink for the powder room.

Eight months after closing, on the way to spending a total of roughly $750,000 on the project, Mr. Miller had achieved his goal of building the house in time for the summer of 2019 — sort of.

By late June 2019, the house was largely complete, but the furniture was still in boxes. And there was no kitchen, because the cabinets were delayed. Intent on getting the place ready for his family to move in for the Fourth of July weekend, Mr. Miller drove out a few days early and took matters into his own hands.

“He was sleeping on the floor in a sleeping bag,” Ms. Miller said, “and pulling these all-nighters, putting the furniture together.”

The furnishings Mr. Miller screwed together and arranged in the house included a mix of affordable pieces from retailers like Ikea, Pottery Barn and West Elm, as well as a few, more expensive statement pieces the couple had long coveted, like a CH25 lounge chair by Hans J. Wegner for the living room.

While waiting for the kitchen, which would be installed that August, they cooked most of their meals outside on the grill and washed dishes with the garden hose.

“We entertain a lot, and had people here,” Ms. Miller said. “The nice part of not having a kitchen is that you learn to simplify your meals.”

Although they imagined doing more elaborate entertaining in 2020, the house has served other functions since the pandemic struck in March, when the family settled in for a long-term stay.

“It’s been a law firm, an architectural firm, a nursery school and an elementary school,” Ms. Miller said, adding that the family plans to return to Manhattan in September. “This house has been such a savior for us.”

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