Rinse whiting fillets in cold water and pat dry with paper towels. Season both sides of each fillet with salt and pepper. Place the flour in a shallow dish; gently press each fillet into the flour to coat and shake off the excess flour.
Heat the olive oil in a frypan over medium-high heat; cook the fish in the hot oil until the fish flakes easily with a fork, about 4 minutes per side. Brush the melted butter onto the fish in the last minute before removing from the frypan. Serve immediately.