Two Audis and a Volkswagen Golf were taken from the house in Whytbank Row, Clovenfords, between 10pm on Friday, February 22, and 1.35am the following day.
After entry was forced to the house, keys to a grey Volkswagen Golf, black Audi RS4 and silver Audi A4 Allroad were taken, and all three cars were stolen.
The Golf was recovered in Penicuik a short time later.
The Audi RS4, with registration number OE07NMA, was recovered in the Millerhill area of Edinburgh at around 10.25am on Sunday, but it is believed it was first used during a robbery at the Community Hub in Humbie, East Lothian.
That break-in took place around 1.20am on Sunday, February 24, and resulted in the theft of a safe containing a four-figure sum of cash.
The Audi A4 Allroad, with registration number P33 OFT, has not yet been found and police fear it will be used to commit a further crime.
Detective Chief Inspector Paul Batten, of Lothians and Scottish Borders police’s criminal investigation department, said: “The theft of these vehicles, and the the subsequent break-in to the property in East Lothian, are being investigated as part of Operation Greenbay, and this incident is being linked with similar house break-ins and car thefts that have occurred across the east of Scotland over the past month.
“We believe those responsible will look to utilise the vehicles to commit further offences, and we would ask anyone who has seen them since the early hours of Saturday morning to contact police immediately.
“We are also keen to hear from anyone who has any relevant information that can help us identify and trace the culprits.”
Anyone with information can contact police via 101 or Crimestoppers on 0800 555111, quoting incident number 427 of February 22 or incident number 419 of February 23.