- Light your charcoal fire, or set your gas grill to medium-high.
- In a food processor, pulse the butter with the tarragon, hot sauce, salt, and pepper until blended.
- Transfer the tarragon butter to a sheet of plastic wrap and roll it into a 5cm thick log. Refrigerate the butter until slightly firm, about 15 minutes. Slice the butter into 36 pats. (This step can be done up to 5 days before grilling the oysters.)
- 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 immediately.