WATKINS GLEN, N.Y.—On a verdant compound in upstate New York, some 800 fugitives while away their days, basking in the sun while a 24-hour staff caters to their needs.

They’re escapees from a city slaughterhouse. On one grassy spot stands young Frank Lee, a steer who fled through the streets of Brooklyn before being corralled, and now relaxes in the company of other bovine escapees.

The...