File descriptor leak in new integration

Over ~6 hours after a deploy, your service slowly degrades: new client connections start getting refused, logs show Too many open files and accept: too many open files, and the process's open file-descriptor count (from /proc/<pid>/fd) has climbed steadily to its 65536 ulimit -n ceiling and plateaued. Request rate is flat and normal, this isn't a traffic event. Restarting an instance fixes it for a few hours, then it recurs. The deploy added a new outbound integration that makes HTTP calls. How do you triage and find the leak?