For the Tartar Sauce: Combine all of the ingredients in a small bowl. Cover and refrigerate until ready to use.
Heat oil in a deep fryer to 350°F, in a deep cast iron skillet, or in a Dutch oven over medium heat to a depth of about 2-3 inches, until it reaches 350°F. If you don’t have a deep-fry thermometer, sprinkle a few drops of water on the hot oil. If it sizzles, the oil is hot enough.
Drain the oysters in a colander and pat dry with paper towels. Use a toothpick to poke a small hole in the large round part of each oyster.
Put the buttermilk in a shallow dish. In a separate shallow dish, combine the cornmeal, flour, salt, Old Bay seasoning and pepper.
Add the oysters to the buttermilk and soak for a few minutes. Remove the oysters from the buttermilk, gently shake off excess liquid, and dredge in cornmeal mixture to coat on all sides. Shake off excess breading and transfer to a wire rack.
Working in batches so that you don’t overcrowd the pan, fry oysters until golden and crispy, about 2-3 minutes. Drain on paper towels, season with additional salt and pepper (if necessary) while they’re still warm, and serve hot with tartar sauce or other dipping sauce.