vagrant@saio:~$ swift stat -v test test | egrep -i "(timestamp|url)"
                   URL: http://saio:8080/v1/AUTH_test/test/test
           X-Timestamp: 1643914310.47773
vagrant@saio:~$ tail /var/log/syslog | grep "object-"
Feb  3 18:54:37 saio object-6020: STDERR: (6752) accepted ('127.0.0.1', 56050)
Feb  3 18:54:37 saio object-6020: 127.0.0.1 - - [03/Feb/2022:18:54:37 +0000] "HEAD /sdb2/40/AUTH_test/test/test" 200 8 "HEAD http://saio:8080/v1/AUTH_test/test/test" "tx1cc84ff96e4b40d18bdb0-0061fc24ed" "proxy-server 6480" 0.0006 "-" 6752 0
Feb  3 18:54:37 saio object-6020: STDERR: 127.0.0.1 - - [03/Feb/2022 18:54:37] "HEAD /sdb2/40/AUTH_test/test/test HTTP/1.1" 200 423 0.001149 (txn: tx1cc84ff96e4b40d18bdb0-0061fc24ed)
vagrant@saio:~$ swift-get-nodes /etc/swift/object.ring.gz AUTH_test test test | grep /srv/node
such as "export DEVICE=/srv/node"
ssh 127.0.0.2 "ls -lah ${DEVICE:-/srv/node*}/sdb2/objects/40/35a/a161102fba1710ef912af194b8d4635a"
ssh 127.0.0.3 "ls -lah ${DEVICE:-/srv/node*}/sdb3/objects/40/35a/a161102fba1710ef912af194b8d4635a"
ssh 127.0.0.4 "ls -lah ${DEVICE:-/srv/node*}/sdb4/objects/40/35a/a161102fba1710ef912af194b8d4635a"
ssh 127.0.0.1 "ls -lah ${DEVICE:-/srv/node*}/sdb1/objects/40/35a/a161102fba1710ef912af194b8d4635a" # [Handoff]
note: `/srv/node*` is used as default value of `devices`, the real value is set in the config file on each storage node.
vagrant@saio:~$ ls /srv/node2/sdb2/objects/40/35a/a161102fba1710ef912af194b8d4635a/
1643914310.47773.data

...  so you could look at how that does it:

If you're doing something more sophisticated (maybe in the HPC space with direct/non-proxy access) - checkout the "ListEndpoints" middleware: