Rinse the fish and pat dry with paper towels. Lay the fish on a large piece of heavy-duty aluminum foil. Put half the lemon slices inside the fish's cavity; place the remainder on top. Fold the foil to encase the fish, securing the ends sono liquid escapes during baking. Place the package in a large roasting pan.
Bake for about 45 minutes, until the fish flakes easily.
While the fish bakes, prepare the sauce. Melt the butter in a med saucepan over med heat. Stir in the flour to form a smooth paste. Cook for about 2 minutes, then slowly stir in the milk and continue cooking, stirring constantly, until the sauce thickens, about 2 minutes more. Add the parsley, eggs, and salt and pepper to taste.
Divide the salmon into serving portions and spoon the sauce over each before serving.