In a wide, shallow bowl or platter, spread flour in an even layer. Line a rimmed baking sheet with parchment paper. Season fish fillets all over with salt. Working with one piece of fish at a time, dredge fillets in flour, shaking off excess. Transfer to prepared baking sheet.

Working in two or more batches so that you can comfortably fit fillets in one layer without crowding, heat 2 tablespoons (30g) clarified butter in a large nonstick skillet over medium-high heat until shimmering. Add fish, presentation side down, and cook until well browned on on first side, 3 to 4 minutes. Using two spatulas, if necessary, carefully flip each fillet and continue cooking until just cooked through and flesh is opaque throughout, 30 seconds to 1 minute. Carefully transfer fish, well-browned side up, to a serving platter or plates and keep warm. Repeat with remaining fillets and clarified butter, using 2 tablespoons of clarified butter per batch.

Wipe out skillet. Add unsalted butter to skillet, set over medium-high heat and cook until butter is melted and foamy. Continue to cook, swirling pan, until milk solids in butter turn a deep hazelnut brown, 1 to 2 minutes longer.

While butter is browning, squeeze lemon juice all over fish, then sprinkle all over with parsley.

When butter is browned, immediately spoon it all over the fish; it should bubble and foam as soon as it hits the fish. Serve immediately with additional lemon on the side.