Combine the red and white onion with 2 tablespoons extra-virgin olive oil, 1 tablespoon red wine vinegar, 1/4 teaspoon salt, and several grinds of black pepper in a bowl.
Using your hands, mix and squeeze everything together for 5 minutes. (Don't rush through this part; the onions need time to release their juices and mellow.)
Work in the cans of tuna in olive oil, including the oil. Season with lemon juice, additional red wine vinegar and salt. Add pepper, to taste.
Cut Kaiser rolls in half crosswise and layer each bottom half with lettuce leaves. Use a couple of varieties (whatever looks good) for the difference in texture.
On top of each lettuce layer, arrange 4 tomato slices (skip this step if you don't have really tomatoes on hand). Then add about half the tuna mixture and 3 to 4 slices of hard-boiled egg.
Crisscross each sandwich with 2 drained anchovy fillets and strew with Nicoise olives (pit them if desired), chopped scallions, and sliced radishes. Then drizzle generously with more olive oil and top with the remaining Kaiser halves, pressing gently but firmly.