For the fish: Set oven to 200 degrees, and place a heatproof plate in the oven.
Place flour in a baking dish. Season dry fish fillets with salt and pepper, set aside for 5 minutes or until moisture shows on the surface.
Coat fish with flour, shake off any excess, and set aside on a baking sheet.
In a large (12 inch) non-stick skillet, heat 1 tablespoon of the oil over high heat until shimmering.
Add 1 tablespoon of butter and swirl to coat the bottom of the pan. Add two fillets skin side up, to the skillet, and reduce heat to medium high. Cook the fish without distubing it, about 3 minutes, or until the edges are opaque and the bottom is golden.
Carefully flip the fish (you may need two spatulas to do this without breaking it), and continue to cook for another two minutes or until the thickest part flakes when a toothpick is inserted.
Transfer the fish to the oven to keep warm while you cook the remaining fillets.
Wipe the pan clean, and repeat steps 6 through 11 with the other two fillets.
For the browned butter: Heat remaining 2 tablespoons butter in a stainless steel skillet over medium high heat until the butter melts.
Continue to cook, swirling the pan, until the butter is golden brown and smells nutty, about 1 - 1 1/2 minutes more. Remove from heat.
To serve: Remove fish from the oven and sprinkle with parsley. Add the lemon juice and capers to the browned butter, swirl to combine. Taste butter sauce, add salt or pepper to taste. Spoon the butter sauce over the fish, and serve immediately.