I see you can get shielded Cat6A. Maybe I can ground the shield.
the main difference between cat6a and cat6 is that cat6a is shielded. that shielding is intended to be grounded at both ends, typically via the jack/receptacle. a cat6a termination (male plug) has a metal face on it that connects to the cable shielding. that metal face is what grounds against the jack.
in real performance, cat6a and cat6 are the same, but cat6a buys you some extra insurance. it protects against interference, especially at higher link speeds or longer runs.
in a normal house, i would suggest cat6 instead of cat6a because it's easier to work with. the cable is thinner and more flexible. further, the run lengths in a normal house usually don't get closet to the limits, so the shielding kinda doesn't help.
however, if you want to run copper between buildings, i say just 'go big' and do cat6a.
as mentioned, there are advantages to fiber, though. if it were me personally, i would go fiber no question ("go big!"), but i assumed you didn't want to shop for fiber networking equipment and learn about how that stuff works. i was more imagining you would just plug into your existing, ISP-provided wifi router or something and get some cheap switch in the other building.