Preheat a panini press. In a medium bowl, flake the tuna. Stir in 1/4 cup olive oil, the lemon peel, lemon juice, onion, celery and sun-dried tomatoes; season with salt and pepper.
Brush each slice of bread on one side with olive oil and place oiled side down on a work surface. Divide the tuna salad among 4 slices, making a well in the center. Top the tuna salad with the pepperoncini.
In a small nonstick skillet, heat the remaining 1/4 cup olive oil over high heat. One at a time, fry the eggs just until the egg whites set, about 1 minute each. Set one egg sunny-side up on top of the tuna and pepperoncini, aligning the yolk with the well. Set the remaining bread slices in place, oiled side up.
Working in batches, grill the sandwiches in the panini press until the bread is crisp and golden, about 1 minute.