Use 1 table spoon of tarragon. It can over power.
Liked these grilled oysters - but -- not many of them opened and if left on the grill longer than 5 minutes the oysters lost their juiciness. I had to take them off grill at 5 minutes and open with an oyster knife, then place back out on the grill to melt the tarragon/butter. The oysters I purchased were the freshest possible from the fishmonger that day.