Wash the mussels and clean them well; discard any open or broken shells. Pour the wine into a deep frying pan or pot, add the mussels and put them to boil. Then reduce the heat and let them simmer for 2-3 minutes, until the mussels open. Those that do not open need to be discarded.
Then remove the mussels from the pan and keep them warm. Strain the wine, return it to the pan, allow it to boil over low heat and add the butter, stirring until dissolved. Add the mussels and leave them to warm up completely for 1-2 minutes. Peel the potatoes and cut them into sticks measuring 3″ (8 cm) in length and with a thickness of 1/2″ (1 cm).
Place them in a bowl with cold water for 1 hour, then remove and place them on paper towels to drain them. Heat oil in a deep fryer or a deep frying pan to 356°F (180 °C). Put the potatoes to fry in small portions for about 4-5 min., remove them when they start to gain a slight golden color and leave them on the paper towels to drain.
Heat the oil to 374°F (190 °C), fry the potatoes again for about 2 minutes until they are crispy and golden. Remove and put them to drain again, then sprinkle them with salt. Distribute the mussels together with the liquid in which they were cooked into 4 deep plates, sprinkle them with the remaining parsley and put a quarter lemon in each plate.
Serve with an additional plate for the shells and a bowl of water and lemon juice. Serve the potatoes in a separate dish with a small bowl of mayonnaise.