Prepare the fish by removing the skin if necessary, (optional). Then lightly coat all sides with olive oil and salt and pepper.
Place the flour in a bowl and dredge the fish pieces to coat.
Place a large skillet on the stove on medium high heat and add 1 tablespoon of olive oil and 1 tablespoon of butter. Fry the fish on the first side until about a third of the fish becomes opaque, (2 minutes). Flip and fry the second side until the sides of the fish filet or nearly completely opaque, (2 minutes). Remove the fish from the skillet.
Wipe the skillet with paper towels to remove most of the burnt flour and then add 2 tablespoons olive oil and 1 tablespoon butter on medium heat. Stir in a 4 teaspoons of flour and cook for 1 minute. (Using leftover dredging flour would be okay).
Whisk in the broth to combine with the flour and stir until lump free. Then add the lemon juice and capers. Bring to a slow boil and stir to thicken, 2 minutes.
Lower the heat to low and add the fish back to the skillet along with 2 tablespoons chopped butter. Spoon the sauce over the fish to heat back up and to melt the added butter.
Taste the sauce and add salt and pepper as desired. Serve hot with the sauce and the side dish of your choice.