Stir together butter, tarragon, hot sauce, salt, and pepper in a small bowl until evenly blended. Transfer tarragon butter to a sheet of plastic wrap, and roll into a 2- x 8-inch log. Refrigerate butter until firm enough to slice, 15 to 30 minutes. Cut into 36 slices.

Preheat grill to medium-high (400°F to 450°F). Working in 2 batches, place oysters, flat sides up, on ungreased grates. Grill, covered, until oysters open, 2 to 4 minutes. Using tongs, transfer oysters to a platter, making sure to keep liquor inside shells. Quickly remove and discard top shells with tongs or heatproof gloves; loosen oysters from bottom shells. Top each oyster with a slice of tarragon butter. Return oysters to grill, and cook, uncovered, until butter is melted and oysters are hot, 30 seconds to 1 minute. Repeat with remaining oysters. Serve immediately.