Rinse the sole fillets under cold water and pat them dry with paper towels.

In a shallow dish, combine the flour, salt, and black pepper. Dredge each sole fillet in the flour mixture, shaking off any excess.

In a large skillet, melt 2 tablespoons of butter over medium heat. Add the sole fillets and cook for about 3-4 minutes on each side until golden brown.

Remove the fillets from the skillet and place them on a warm plate.

In the same skillet, add the remaining 2 tablespoons of butter and let it melt, cooking until it turns light golden brown.

Remove from heat and stir in the lemon juice and chopped parsley. Pour the brown butter sauce over the cooked sole fillets.

Serve immediately with lemon wedges on the side.