*I* don't like seafood but my family does so each New Year in an untraditional Italian way, I make a seafood feast. Oysters are a favorite as is tarragon. I found a recipe all over the internet & adjusted it to how Mom would like it best. I will make tarragon chicken for myself. ;)
Combine butter & tarragon in a small bowl. Season with salt and pepper.
Place the oysters on the hot grill, flat-side up. Cover the grill and cook until the oysters open, about 5 minutes. Using tongs, transfer the oysters to a platter, trying to keep the liquor inside.
Quickly remove the top shells and loosen the oysters from the bottom shells. Top each oyster with a pat of tarragon butter and return the oysters to the grill. Cover the grill and cook until the butter is mostly melted and the oysters are hot, about 1 minute. Serve right away.