Soak a generous handful of Hickory Wood Chips in water. Mix a few handfuls of Hickory Wood Chips (unsoaked) in the charcoal. Fully open the draft door on the ceramic base of the Big Green Egg, distribute three fire starters over the charcoal and light them. Leave the lid of the EGG open for 10-12 minutes.
In the meantime, remove the veal from the refrigerator. Mix together all of the ingredients for the rub and generously rub the mixture on all sides of the meat.
Sprinkle the soaked Wood Chips over the glowing charcoal. Set up the convEGGtor and place the grid inside the EGG. Place the veal on the grid, close the lid of the EGG and heat to a temperature of approx. 100°C. Leave the veal to cook for about 50 minutes until the meat has reached a core temperature of 52°C. You can measure the temperature with the Instant Read Digital Thermometer or Dual Probe Remote Thermometer.
In the meantime, make the tuna sauce. Drain the tuna and the capers and in the blender, mix with the other ingredients for the sauce. The structure of this sauce can be slightly coarse. Season the sauce with salt and pepper to taste and cover to store in the refrigerator until ready to serve.
Remove the veal from the EGG and allow to cool. Wrap in plastic and store in the refrigerator until ready to serve.
Cut the veal into nice thin slices before serving the vitello tonnato. Drain the caper fruits and capers and cut the caper fruits in half lengthwise. Distribute the mixed lettuce over the plates and place the sliced veal on top. Garnish with the tuna sauce, the whole and halved caper fruits and the capers, and then grate the zest from one half lemon on top.