Preheat an outdoor grill or griddle pan to medium-high heat. In a large, shallow dish, whisk together about 1/3 cup EVOO, the garlic, anchovy paste and rosemary; season with pepper. Add the tuna, turn to coat and thread on metal skewers. Let marinate for a few minutes for the flavors to soak into the fish. Place the tuna on the grill or griddle and cook, turning a quarter turn each minute or so, for 5 minutes.
While the tuna grills, char the bread for grill marks or broil to char at the edges.
Coarsely chop the olives and cut the caperberries into long, thin wedges, or run your knife through the drained capers. Pile half of the tuna on each ciabatta loaf bottom and top with the olives, caperberries, onion, parsley and arugula. Dress each sandwich with the lemon juice and a drizzle of EVOO; season with pepper and top with the eggs. Set the ciabatta tops into place and cut each loaf into 4 pieces (serve half a loaf, or 2 pieces, per person).