1. Cut the fish fillets into fat strips, about 4 x 10cm. Put the flour in a bowl and season; put the eggs in another bowl. In a third bowl, combine the breadcrumbs, parmigiano reggiano and parsley, then season. Dip each piece of fish into the flour, then the egg, then coat in breadcrumbs. This can be done in advance and the fish kept in the fridge for up to 2 hours.
2. Heat the olive oil and butter in a large frying pan over a medium-high heat. Cook the fish in batches for about 2 minutes on each side, until lightly golden, adding a little more oil and butter if necessary.
3. To make the caper mayonnaise, put all the ingredients in a bowl and mix to combine. Serve the fish fingers with the lemon wedges, caper mayonnaise, salad, and potato wedges.