Spread the flour into a shallow bowl.
Season both sides of the fish with salt and black pepper.
Dredge each fillet in the flour, shaking off any excess.
Heat the vegetable oil in a large nonstick skillet over medium-high heat.
Cook the fish for 3–4 minutes on the first side until golden brown.
Carefully flip the fillets and cook for another 30 seconds to 1 minute until the fish flakes easily. Transfer to a serving platter and keep warm.
Wipe out the skillet and return it to medium-high heat.
Add the chicken broth and simmer for about 2 minutes until slightly reduced.
Stir in the capers, 4 tablespoons of butter, and the softened butter mixed with ¼ teaspoon flour.
Whisk continuously until the sauce becomes smooth, creamy, and lightly thickened.
Stir in the fresh lemon juice and chopped parsley.
Taste and adjust seasoning with salt if needed.
Spoon the warm sauce over the fish and serve immediately.