Make a tartar sauce by stirring the mayonnaise, lemon juice, lemon zest, dill pickle, and 1/4 teaspoon cayenne pepper together in a small bowl until thoroughly combined; set aside.
Mix the panko crumbs, 1/8 teaspoon cayenne pepper, salt, and black pepper together in a shallow bowl.
Heat the canola oil in a deep-fryer or saucepan to 360 degrees F (180 degrees C). Measure temperature with a deep frying thermometer for accuracy.
Dip an oyster into the beaten egg, and press into panko crumbs to coat. Gently toss between your hands so any bread crumbs that haven't stuck can fall away. Place the breaded oysters onto a plate while breading the rest; do not stack.
Cook the oysters, two at a time, in the hot oil until the panko is browned, about 1 minute per side. Transfer the fried oysters to a baking sheet lined with paper towels, and place in oven to keep warm while frying remaining oysters. Serve with the tartar sauce for dipping.