How to monitor all outgoing http calls from my debian linux server?
I assume you have root priviledges on your production server.
You can try using tcpdump, log the output to file and then parse/filter it as you see fit with tools like grep, sed etc...
Please note that tcpdump includes a failrly complex and complete set of filtering options, so be sure to check the man page.
This is a sample command to log all outgoing https requests (run as root):
tcpdump -s 0 'tcp dst port 443'