Preheat the oven to 200°C (400°F/Gas 6). Lightly grease a baking tray.
Melt the butter in a small frying pan and cook the onion over low heat for 5 minutes, or until soft but not brown. Combine the onion, tuna, capers, parsley and parmesan in a bowl and season with salt and pepper.
Cut the filo pastry sheets in half widthways. Layer four of the half sheets together, brushing each with melted butter. Keep the remaining pastry covered with a damp tea towel. Spoon half the tuna mixture onto one end of the buttered pastry, leaving a border. Make a well in the centre of the mixture and break an egg into the well, being careful to leave it whole.
Layer two more sheets of filo together, brushing with melted butter, and place on top of the tuna and egg. Fold in the pastry sides, then carefully roll into a firm parcel, keeping the egg whole. Place on the prepared tray and brush with melted butter. Repeat with the remaining pastry, filling and egg. Bake for 15 minutes, or until the pastry is golden brown. Serve warm or at room temperature with a green salad.