Hi, go to HPE QuickSpecs portal and, under the Online QuickSpecs section, select Search All QuickSpecs.
Once there search for the string "HP ProLiant ML350p Generation 8 (Gen8)".
The (worldwide valid) HPE QuickSpecs for your HP ProLiant ML350p Generation 8 (Gen8) server is the Edition 57 published 27 August 2015 (Retired).
There you will find that supported and certified HP Network Interface Cards (with 1G and 10G, SFP/SFP+ and RJ45) are listed at page 34:
If I were you - to be 100% sure the NIC will be supported - I'll search one of those 10G Base-T adapters listed above.
Since you're looking for 10GBase-T (Copper RJ45) port(s) - you wrote about Cat. 6 cabling (which is OK) - suitable NICs (both dual ports) would be:
- HPE Ethernet 10Gb 2-port 530T Adapter (HP SKU: 656596-B21)
- HPE Ethernet 10Gb 2-port 561T Adapter (HP SKU: 716591-B21)