Place a cooling rack over a baking sheet and put in an oven set to WARM, or no more than 200ºF. Warm 4 dinner plates.

Season the flounder on both sides with salt and pepper. Combine flour and cornstarch in a shallow bowl. Whip eggs until no strands of white remain and put in another shallow bowl.

Place a large nonstick skillet over medium-high heat. When hot, add 2 teaspoons olive oil and heat; when it just begins to shimmer, add 2 teaspoons butter.

Keeping an eye on the butter, quickly dip 2 of the fillets in flour for a light coat, then dip into egg for a coat; allow excess egg to drip off, then slip fish into hot skillet. Immediately reduce the heat to medium-low.

Cook until nicely browned on the bottom, about 5 minutes. Turn carefully with a fish spatula and season the browned side with a light sprinkle of paprika and a touch of cayenne. Cook until fish is opaque in the center, 3 to 4 minutes. Transfer to the rack in the heated oven.

Wipe out the pan and repeat the process with remaining fish.

To finish, wipe out the skillet, return to low heat, and add the remaining 6 tablespoons of butter, the capers, parsley, and lemon juice. Cook, swirling the pan, until the butter melts. Plate the fish and divide the Lemon Caper Butter Sauce over the fillets. Serve at once.