Get a used Mellanox X3 card (warning! there are different versions with 8x2.0 or 4x3.0 PCIe slot demand! look at your mobo which one will fit. DONT run these cards with less than 4 or 8 lanes!!!).
Microtik has CAT modules for 10G speed in stock, they are rather expensive and, worst of all, they get seriously HOT!. But if you run only one of them, this should work. But keep an eye on the temperature, if it gets too high, the switch turns the port OFF...
BTW, your CAT6 cable is not the problem, the plugs on both ends are! Usually they are NOT CAT6, therefor they can produce a lot of problems. Also, lenghts >10m can be problematic! Be prepared for Link losses, Line Drops and Port resets... you have been warned.
Better go for fiber or Direct Attach if possible.