Heat oven to 200 degrees and place a large oven-safe plate or baking sheet inside.
Place flour on a large, shallow plate. Season both sides of fish fillets with salt and pepper to taste. Dredge fish in flour, shaking off excess.
In a 12-inch nonstick or enamel-lined skillet over medium-high heat, heat 2 tablespoons clarified butter until bubbling. Place half of the fish fillets in the pan and cook until just done, 2 to 3 minutes per side, then transfer to the plate or baking sheet in the oven to keep warm. Add 2 more tablespoons clarified butter to skillet and heat until bubbling, then cook remaining fillets. Wipe out the skillet.
Arrange the fish on a warm serving platter. Top with parsley. In reserved skillet, heat remaining 4 tablespoons unsalted butter until bubbling and golden, 1 to 2 minutes, then pour evenly over fillets. Serve immediately, with lemon wedges on the side.
½ cup all-purpose flour
6 4-ounce skinless, boneless sole or other thin fish fillets, patted dry
Kosher salt, to taste
Freshly ground white or black pepper, to taste
4 tablespoons clarified butter
4 tablespoons unsalted butter, diced, at room temperature
3 tablespoons minced parsley
1 lemon, cut into wedges, for serving