Make the tartar sauce: Combine all of the tartar sauce ingredients in a small bowl. Cover and refrigerate until ready to serve.
Drain and dry the oysters: Pour the oysters into a colander to drain, rinse briefly, and pat dry with paper towels. Get them as dry as possible.
Poke each oyster: Use a toothpick to poke a small hole in the large round part of each oyster to release steam and pressure during frying.
Set up the dredging station: Put the buttermilk in a shallow dish. In a separate shallow dish, whisk together the cornmeal, flour, salt, Old Bay seasoning, and pepper.
Soak the oysters: Add the oysters to the buttermilk and soak for a few minutes.
Dredge and rest: Remove the oysters from the buttermilk, gently shake off the excess liquid, and dredge in the cornmeal mixture to coat all sides. Shake off the loose breading and transfer to a wire rack to rest while the oil heats.
Heat the oil: Heat oil in a deep fryer, a deep cast iron skillet, or a Dutch oven over medium heat to a depth of about 2 to 3 inches, until it reaches 350°F. If you don't have a thermometer, sprinkle a few drops of water on the hot oil. If it sizzles, the oil is ready.
Fry in batches: Working in batches so you don't overcrowd the pan, fry the oysters until deep golden and crisp, about 2 to 3 minutes. Adjust the heat to keep the oil between 350°F and 375°F.
Drain and serve: Drain on a wire rack or paper towels, season with additional salt and pepper while still warm, and serve hot with tartar sauce.