Blanch basil leaves in a saucepan of boiling water for 30 seconds and drain. Transfer leaves to a medium bowl, filled with ice water, to cool. Drain and pat leaves dry. Finely chop basil. Mix mayonnaise, mustard, lemon juice and cayenne in a medium bowl. Set aside 1/2 cup dressing for crab cakes. Mix basil with remaining mayonnaise dressing.

Heat 2 tbsp olive oil in a heavy large skillet over medium heat. Add celery and onion then saute until tender, about 5 minutes. Transfer to a large bowl, stir in crabmeat, 2/3 cup bread crumbs, chopped chives, parsley and reserved 1/2 cup mayonnaise dressing. Season mixture to taste with salt and pepper. Mix very gently so as not to crush crab meat. Using a scant 1/3 cup mixture for each cake, form into twelve 2 1/2 inch diameter cakes. Arrange cakes on baking sheet and chill for at least one hour. Place flour in a small bowl. In another small bowl, whisk eggs to blend. Place remaining 2 cups bread crumbs in a medium bowl. Coat each crab cake with flour. Dip into beaten egg, then into bread crumbs, coating completely. Heat 1 tbsp vegetable oil in large skillet over medium heat. Add crab cakes in batches and cook until golden brown and heated through, adding more oil as necessary, about 5 minutes per side. Serve with basil dressing.