Put the mayonnaise in a medium bowl, add the mustard, capers, cornichons anchovy, herbs, shallot, paprika, hot sauce, vinegar, lemon zest, and lemon juice and stir to blend. Taste and adjust the seasoning; set aside.
Heat the oil in a large heavy saucepan over high heat until it registers 350°F on a deep-frying thermometer.
Meanwhile, combine the cornmeal, flour, salt and pepper, and cayenne in a shallow bowl. Put the buttermilk in another shallow bowl and add the oysters. One by one, transfer the oysters from the buttermilk to the cornmeal mixture, turning to coat thoroughly, and place the oysters on a rack set over a small baking sheet.
Place a spoonful of the rémoulade in each clean oyster shell and arrange on a serving platter.
Carefully drop the oysters, in batches of about 6 at a time, into the hot oil and cook, turning once or twice, until golden, 2 to 3 minutes. Transfer to paper towels to drain. Place the fried oysters in the shells and serve.