Make the aioli and begin marinating the tuna early in the day. Microwave the lemon and lime in a small bowl to release the essential oils and set aside just long enough to be able to handle. Meanwhile, through the feed tube of a running blender, add 1 at a time, the egg yolks, dry mustard, salt, and white pepper. By this time, the micro-waved lemon and lime should be cool enough to handle. Leaving the blender running, squeeze the juice from half of 1 lemon through the feed opening, then add the oil in a slow stream. Remove to a bowl and refrigerate.
Season the tuna steaks with salt and pepper, place them in a nonreactive container and squeeze the rest of the lemon and the lime over them. Cover and refrigerate for 2 hours.
Heat the grill or preheat the broiler in the oven. Brush the marinated tuna with 1 tablespoon or so of olive oil. Cook until browned, until fish flakes with a fork, 3 to 6 minutes per side, and then let rest. Leave grill hot or oven on to toast bread.
Brush olive oil (about 1 tablespoon per half) on Italian bread and toast lightly, either on the grill or under broiler.
Flake the tuna, place in a bowl with scallions and fennel and fold in the aioli. Spoon tuna mixture onto the toasted Italian bread, drizzle a small amount of balsamic vinegar over, and top with Cheddar. Heat briefly in the oven to melt cheese. Garnish with fresh dill. This is great served with fresh tomato soup, if desired.
Recipe courtesy of Robert Irvine with Brian O'Reilly, HarperCollins Publishers, copyright 2007.