Put the tuna in a bowl, break it up with a spoon, then mix in the mayo, onion, a quarter-teaspoon of salt and a few grinds of black pepper.
Put all the tapenade ingredients in a food processor and blitz to a chunky paste.
Put the 100ml olive oil for the potatoes in a large, nonstick frying pan on a medium-high heat, then fry the potato slices, flipping often, for 10-12 minutes, until golden brown on both sides.
Use a slotted spoon to transfer the potatoes to a plate lined with kitchen paper, to drain, then season with a half-teaspoon of salt.
To assemble the toasties, spread each slice of bread with a quarter of the tapenade, then top with a 50g sprinkling of grated mozzarella.
Divide the tuna mix between two of the slices of bread, then top that with the fried potatoes.
Put the parsley leaves on top of the potatoes, then carefully close each sandwich by putting the two tapenade and cheese-topped bread slices on top.
Return the frying pan to a medium heat, lay one of the sandwiches in the pan and fry for about six minutes, flipping it once halfway through, until golden and toasted on each side, and with melted cheese inside.
Remove from the pan and drain on a rack or plate lined with kitchen paper, then repeat with the second sandwich.
Slice each sandwich in half, and serve immediately.