Place cleaned crabmeat in a medium mixing bowl. Pour the egg white over crabmeat slowly, stopping occasionally to mix it through. When the crabmeat has absorbed the egg white and feels slightly sticky to the touch, sift the flour over the mixture and sprinkle with chives, black pepper, cayenne, and 1 teaspoons salt evenly over the top. Lift the crabmeat from the bottom of the bowl, turning it over gently, to mix ingredients without overhandling.
Separate the mixture into 8 equal portions and gently roll each between the flattened palms of your hands to form loose balls. Flatten slightly and transfer to a plate. Sprinkle both sides liberally with the remaining 1 teaspoons of salt and refrigerate for at least one hour before cooking.
Line a baking sheet with paper towels. Fry the crab cakes in two batches to ensure a crisp crust. Using a small (8") heavy skillet, melt half the butter and oil together until the mixture is foamy and begins to brown. Carefully place crab cakes in the hot fat and fry until a crust forms, turning only once, about 2 minutes per side. The fat should be sizzling hot, enabling a crisp crust to form before the crab absorbs the cooking fat. This is the Southern secret to perfect crab cakes. A small pastry spatula (with a thin tongue) will make lifting and turning the delicate crab cakes a lot easier. Remove the crab cakes and drain in the prepared pan. Cover loosely with aluminum foil to keep warm while you make the second batch.
Carefully pour off the cooking fat from the first batch, wipe out the pan, and return it to the heat. Prepare the second batch of crab cakes using the remaining butter and oil.