Season the fish fillets with salt and pepper. Make the egg batter by whisking together the eggs and milk in a shallow bowl, along with a pinch of salt.
Put the flour on a plate. Coat each fillet lightly on both sides, then submerge the fish in the egg batter. (This can be done up to 30 minutes ahead.)
Pour about ¼ inch oil in a wide skillet over medium-high heat. When the oil looks wavy, lift the fish from the egg batter one piece at a time, allowing excess batter to drip off. (Discard remaining batter.) Carefully lay each fillet in the pan and let fry gently until golden, about 2 minutes per side. Remove and blot briefly on paper towels, then transfer to a warmed serving platter.
Pour off any remaining oil and return the pan to the heat. Melt the butter in the pan and add the green garlic, and a little salt and pepper. Let cook without browning for about 1 minute, then add the lemon juice.
Spoon the green garlic butter over the fish. Sprinkle with parsley, garnish with lemon wedges and serve immediately.