To make the batter, whisk together eggs, cream, 1/4 cup white wine, 4 tbsp parsley, Parmesan, salt and pepper.
Heat a sauté pan over medium-high heat. Add oil. Dredge the fish in flour and shake off the excess. Soak in egg batter until coated well. Lace fish, skin side up, in sauté pan. Cook 3-5 minutes per side. Remove cooked fillets to another dish until all fillets are done. Hold in a 250° oven while you make the sauce.
Remove any excess oil and debris from sauté pan. Over medium heat, add shallots and 1 cup of wine. Reduce by two-thirds. Add lemon juice and zest while whisking vigorously. Add butter, a piece at a time, until all is incorporated and the sauce is emulsified. Finish with 3 tbsp parsley.