In a large bowl, mix 1 1/2 cups of the bread crumbs with 1/2 cup of the parsley, 1 1/2 teaspoons of salt and 1/2 teaspoon of pepper.
In a small saucepan, melt 2 tablespoons of the butter. Add the shallot and cook over moderate heat until translucent, about 1 minute. Add the chicken stock and lemon juice and boil over high heat until reduced to 1 cup, about 15 minutes.
Whisk the crème fraîche into the sauce along with the remaining 1/2 cup of parsley and 1/4 cup of bread crumbs. Scrape the sauce into a blender and puree. Strain the sauce back into the saucepan and rewarm gently.
In a large skillet, melt the remaining 2 tablespoons of butter in the olive oil over moderate heat. Season the sea bass fillets with salt and pepper, then dredge them in flour, dip in the beaten eggs and coat with the parsley and bread crumb mixture. When the butter starts to brown slightly, add the fillets to the skillet and cook until browned on the bottom, about 3 minutes. Flip the fillets and cook until just white throughout, 2 to 3 minutes longer. Transfer the fish to plates, spoon the sauce alongside and serve.