Preheat grill to 450°. Pulse first 8 ingredients in a food processor until well combined.
Arrange oysters in a single layer on grill. Spoon 2 tsp. butter mixture into each oyster; grill, uncovered, 7 minutes or until edges curl.
Broiled Oysters: Preheat broiler with oven rack 3 inches from heat. Prepare recipe as directed, placing oysters in a single layer in a jelly-roll pan. Broil 4 minutes or until edges curl and butter drips over the shell.