Heat grill to high (450°F to 550°F). Stir together butter, parsley, tarragon, jalapeño, Fresno chile, and ¼ cup of the pecorino Romano in a bowl until thoroughly combined. Let stand at room temperature.

Scrub oysters with a stiff brush under running water. Insert an oyster knife into the hinged edge of each oyster; twist knife handle back and forth until top shell is loose. Slide oyster knife along the bottom of the top shell to detach muscle. Remove and discard top shell, keeping the oyster in the deeper bottom shell.

Place oysters on a rimmed baking sheet. Top each with about ½ tablespoon butter mixture. Transfer oysters directly to grill rack. Grill, covered, until oyster juices are bubbling and butter has melted, about 5 minutes. Top with remaining ¾ cup pecorino Romano, and grill, covered, until cheese has melted, about 1 minute. Remove from grill. Serve with hot sauce and saltines.