If using fresh artichoke bottoms for the sauce, poach until soft in water with a little lemon juice and 1 Tbs. salt. Cut the artichokes into eighths and set aside.
Sprinkle the lemon juice over the fish fillets. Stir the salt and pepper into the flour with a fork, and dredge the fillets in the seasoned flour. Shake off the excess flour, dip the fillets in the beaten eggs, and dredge through the flour again. Knock off the excess flour.
Heat the 4 Tbs. butter over medium-high heat in a large, heavy skillet and sauté until the fish is cooked--about three minutes per side. Remove the fish and keep warm.
To make the sauce, add the white wine to the pan in which you sautéed the fish, and whisk to dissolve the pan juices. Bring to a boil until the wine is reduced by two-thirds. Lower the heat to medium and add all the remaining sauce ingredients except the butter. Cook until the mushrooms no longer break when flexed.
Lower the heat to almost off, and add the butter, a tablespoon at a time, agitating the pan until the butter has blended in completely.
Place the fish on serving plates and top with the sauce.