Scoop all of the watermelon slices into a re-sealable food bag. Add a pinch of salt and chill in the fridge for 2-3 days.
When the watermelon is ready, preheat the oven to 150C/130C Fan/Gas 4.
Toast the bread, then slice each in half widthways to create four slices, each with one toasted side and one untoasted side. Sandwich them between 2 clean baking trays. Bake in the oven for 5 minutes, or until hard and crisp. Remove the trays from the oven and set the bread aside to cool, still sandwiched between the trays.
Lay a sheet of cling film onto a clean work surface. Place the tuna steaks on top and cover with a second sheet of cling film. Flatten the tuna steaks to a thickness of 2mm using a meat mallet or rolling pin. (Do this one steak at a time if it is easier.)
Trim each piece of flattened tuna into a circle, using an upturned bowl as a guide. Set aside.
Chop the tuna trimmings very finely and place them into a bowl. Squeeze over half of the lemon juice and add the chives and shallots. Season, to taste, with salt and freshly ground black pepper, then mix well.
Heat the butter in a frying pan over a high heat until it starts to brown. Chop the watermelon slices into small pieces and add them to the pan in batches, frying for 1-2 minutes on each side, or until caramelised and heated through. Season with a pinch of salt. Remove from the pan and set aside on a warmed plate while you fry the remaining watermelon.
To serve, spoon some of the tuna tartare onto each piece of toast. Place onto the serving plates and top with a piece of cooked watermelon. Turn over each circle of flattened tuna carpaccio, discarding the cling film, and place one circle alongside each toast. Brush each circle of tuna carpaccio with a little of the extra virgin olive oil, then sprinkle over the shallot and chives, squeeze over the lemon juice, and season, to taste, with freshly ground black pepper.