Peel the egg and place it in your food processor with the lemon juice, mustard and a third of the tuna. Blitz until smooth.
Add a tbsp of the olives’ brine and then with the machine running add 4 tbsp of olive oil, one at a time. Add the rest of the tuna to the filling and barely pulse to combine.
Cut the tomatoes in half and scoop out some of the insides. Chop the tomato insides and the olives, stir into the filling with black pepper and the basil, first torn into pieces.
Season the tomatoes with a tiny sprinkle of salt, and then fill.