Special equipment: a deep-fry thermometer; toothpicks
Heat 3 inches of oil in a medium saucepan over medium heat until the oil reaches 350 degrees F.
Meanwhile, stir together the cabbage, carrots, vinegar, 1/4 teaspoon salt and 2 tablespoons of the mayonnaise in a medium bowl. Set aside.
Heat a cast-iron griddle or large skillet over medium-high heat. Cut the slider buns in half and place them cut-side down on the griddle. Toast until golden brown, 2 to 3 minutes. Remove and set aside to cool.
In a medium bowl, whisk together the cornmeal, rice flour, basil, garlic powder, oregano, paprika, dry mustard, onion powder, white pepper, thyme and 1 teaspoon salt.
Place the beaten eggs in a small bowl. Dip an oyster into the egg, then dredge it in the seasoned cornmeal. Return the cornmeal-crusted oyster to the egg and then dredge one more time in the cornmeal. Repeat with the remaining oysters.
Carefully place the oysters into the hot oil and fry until golden, about 3 minutes per side. Remove to a paper-towel-lined plate to drain.
To assemble the sandwiches: Spread the cut-sides of the buns with the remaining 4 tablespoons mayonnaise. Place a dollop of ketchup on the bottom buns, set a fried oyster on top of the ketchup, and top with 1 heaping tablespoon coleslaw. Cover with the top buns. Secure with a toothpick.