In a medium bowl, add the fish and squeeze the juice from the lemon on it. Add coldest tap water to cover the fish. Let stand for 15 minutes.
Rinse the fish and use paper towel to dry it. Season the fish on both sides by drizzling it with one tablespoon olive oil and sprinkle the salt and pepper on both sides.
Over medium-high flame, heat 2 tablespoons olive oil. in a non-stick frying pan.Note: Non-stick pan is important so that the fish does not stick.
Fry the fish, one to two pieces at a time. Cook for three minutes per side. It should be lightly browned. Remove to a serving plate and set aside while you cook the remaining fish.Note: Use a fish turner/spatula and handle the fish carefully. This is a very tender fish and it will flake apart easily. Do not be tempted to over cook it as it will be dry and chewy.
MAKE THE SAUCE:
In the same frying pan used to cook the fish, add the lemon slices in a single layer. Cook the lemons on both sides for about one minute total.
Add the capers, chopped parsley, a pinch of salt and pepper to taste. Stir to combine with the lemons. Cook for about one minute.
Add the white wine. Simmer about five minutes until it reduces to half.
Add half of the remaining tablespoon olive oil and use a wooden spoon to incorporate with the mixture. Continue stirring for about one minute.
Add the remaining olive oil and continue the same process of incorporating the olive oil.
Remove from the heat.
Pour the sauce over the cooked flounder filets. Garnish with parsley if desired. Plate and serve.