Special equipment: a deep-fat thermometer
For the wet soak: In a nonreactive bowl or gallon re-sealable bag, combine the buttermilk, beaten egg, hot sauce, basil, granulated garlic and white pepper and mix well. Add the oysters and any liquid from shucking. Cover and refrigerate 15 minutes.
For the breading and frying: In a shallow bowl, combine the panko, cornmeal, flour, dried basil, granulated garlic, cayenne, 1 teaspoons salt, pepper and paprika and mix well.
Heat the oil to 350 degrees F in a large cast-iron skillet.
Drain the oysters from the brine and dredge through the breading mixture, pressing gently to adhere. Carefully add to the hot oil in batches, to not crowd, and cook 3 to 4 minutes per side. Remove to a paper towel-lined plate and season with the remaining 1 to 2 teaspoons sea salt while hot.
For the sandwiches: Preheat the oven to a low broil. Butter the rolls and place on a baking sheet. Broil until golden, 2 to 3 minutes, and remove from oven.
Spread Miss Dixie's Remoulade on both sides of the rolls. Layer on the oysters, lettuce, tomatoes and pickles. Top with other half of bread and serve immediately.
In a food processor, add all ingredients and process until smooth, 25 to 35 seconds depending on blender speed. Scrape down sides as needed. Refrigerate until ready to use.