Rinse the sole fillets under cold water and pat them dry with paper towels. This helps the flour adhere better.

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

In a large skillet, melt 2 tablespoons of butter over medium heat. Once the butter is foamy, add the sole fillets in a single layer. Cook for about 2-3 minutes on each side until golden brown and cooked through. You may need to do this in batches if your skillet is small.

Remove the cooked fillets from the skillet and place them on a serving platter. Keep warm.

In the same skillet, add the remaining 2 tablespoons of butter and let it melt. Cook for about 1-2 minutes until the butter turns a light golden brown. Remove from heat and stir in the lemon juice and chopped parsley.

Pour the brown butter sauce over the sole fillets and serve immediately with lemon wedges on the side.