Preheat the grill to medium-high (450°F to 500°F). Scoop out the soft centers of the roll halves. leaving a 1/2-inch-thick shell. Spread the butter evenly on the cut side of the roll halves. and set aside. Stir together the mayonnaise. D1ion. hot sauce. and lime juice in a small bowl: chill until ready to use.

Pour the oil to a depth of about 3 inches in a medium Dutch oven: heat to 360°F over medium. Meanwhile, place the buttermilk in a medium bowl. Combine the cornmeal. flour. black pepper. cayenne pepper. lime zest. and I teaspoon of the salt in a medium bowl. Dip the oysters in the buttermilk. and dredge 1n the cornmeal mixture. shaking to remove the excess cornmeal mixture.

Add the oysters, in 2 to 3 batches, to the hot oil,. and fry until golden and just done, 2 to 3 minutes. turning over halfway through cooking. Drain the oysters on paper towels; sprinkle with the remaining ¼ teaspoon salt.

Place the rolls, buttered side down, on the grill grate (or in a hot grill pan over medium-high), and grill just until crisp and grill marks appear, about 3 minutes. Spread the mayonnaise mixture evenly on the grilled sides of the rolls. Divide the lettuce, tomatoes, pickles, and fried oysters evenly among the bottom halves of the rolls; cover with the tops.