Line a baking sheet with parchment paper and set aside.

Combine the butter, mayonnaise, lemon juice, extra-virgin olive oil, green onions, chives, and parsley in the bowl of a food processor and process until well blended and slightly thickened. Add the salt, pepper, and cayenne and process for 15 seconds to blend.

Transfer to a medium bowl with the crabmeat and breadcrumbs and fold gently to combine, being careful not to break up the crabmeat lumps.

Form the crabmeat mixture into eight 2½ to 3-inch round cakes, about 2 ½ ounces each, and pack gently but firmly. Place on the prepared baking sheet, cover with plastic wrap, and refrigerate until well chilled, 1 to 2 hours.

Place the flour in a small shallow dish. In a separate bowl, whisk together the egg and the milk to make an egg wash. Place the panko in a third shallow dish.

Dredge each crab cake in the flour mixture, then in the egg wash, and then the panko crumbs, shaking to remove any excess breading. (Note that the crab cake mixture will be slightly wet and should be handled carefully. If the crab cakes become too loose during the breading, they can be refrigerated again until firm.) Set aside.

Heat ¼ cup of the vegetable oil in a large, heavy skillet over medium heat. When the oil is hot, panfry the crab cakes 3 or 4 at a time until golden brown, about 4 ½ minutes per side. Drain on paper towels, and repeat with the remaining oil and crab cakes.

To serve, arrange one or two crab cakes in the center of four or eight plates and top with a dollop of the Creole Remoulade Sauce. Garnish each plate with greens, if desired, and serve immediately.