On a cool, windy day in February, two big white dogs escaped from a well-known nonprofit farm in Westchester County and ended up on a public footpath deep in a New York State park.

They encountered a 10-pound miniature poodle on a leash. The larger dogs attacked, killing the poodle and then severely injuring its owner. Acting on the recommendations of state law, a local judge ordered the dogs to be euthanized.

The disturbing encounter’s aftermath has been considerable. The farm, Stone Barns Center for Food and Agriculture, which is connected to Blue Hill at Stone Barns, a Michelin-starred restaurant in Pocantico Hills, filed an appeal on Thursday to stay the dogs’ euthanization.

A lawyer for the miniature poodle’s owner, Yong Ging Qian, said she was considering a lawsuit against the farm connected to what she said in a hearing in Mount Pleasant Justice Court were her substantial injuries: a mangled hand, several broken ribs, brain hemorrhaging and a mild heart attack.

The dogs — Luna, a Great Pyrenees, and Owyn, an Akbash — were guardian dogs, bred and trained to protect livestock. They had escaped on Feb. 6 from the farm in Westchester County, 15 miles north of New York City. The restaurant, where a meal costs in the range of $400 per person, has earned two Michelin stars for its innovations in farm-to-table cuisine as well as a Michelin green star, awarded to restaurants “at the forefront” of sustainable practices.

The attack has focused attention on Stone Barns and Blue Hill, as former farm employees, breeders and nearby farmers questioned whether it engaged in practices that created a dangerous situation, an accusation that the farm denies.