Season fish with salt and pepper and dredge in flour to coat, shaking off excess. In a large non-stick skillet heat 1 tablespoon butter over moderately high heat until foam subsides and cook fish 3 minutes each side, or until just cooked through.
Reduce heat to moderately low. Cut remaining butter into pieces and add to skillet. Simmer butter until golden brown with a nutty aroma, about 5 minutes. Remove skillet from heat and immediately add lemon juice, salt, parsley, and pepper to taste, swirling skillet to incorporate (butter will foam). Serve sole filets with butter.