That is possible if you have SSH access to the box and you are connecting to domains rather than IP addresses - you need to create new certificates for those domains, that are signed by some CA, and then place them on your box. Since you are the one who initiates the connection, you will be able to decrypt the traffic going and coming between that mining pool and yours.
You also need a second machine that is running a CA which will issue those certificates, as you can't just create any certificate without a CA. You could also use a self-signed certificate, but I am not sure if applications will like that.
Then, you can run Wireshark on the miner and just have it run in the background while you mine and inspect the packets that it dumps.