Preheat a deep-fryer or heavy-bottomed pot, with enough peanut oil to come halfway up the sides of the pot, to 350 degrees F.
Mix all ingredients in a bowl and set aside.
Mix the flour, smoked paprika, cayenne, salt and pepper, to taste, in a shallow bowl. Add the eggs to another bowl and the bread crumbs to a third bowl. Dredge the oysters in the flour mixture, then in egg and then in bread crumbs. Arrange the breaded oysters on a platter lined with parchment. Fry the oysters, in batches, until crisp and light brown, about 1 to 2 minutes Remove to a paper towel lined platter and season with salt and pepper and a dash of lemon juice.
To assemble sandwiches:
Spread mayonnaise and relish on the bottoms of the rolls. Add 4 to 5 oysters to each roll and top with a few leaves of arugula. Cover with the roll tops and serve.