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.)
Cut kaiser rolls in half crosswise and spread 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 good — really good — 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 niçoise olives (pit them if desired), chopped scallion greens, and sliced radishes.
Then drizzle generously with more olive oil and top with the remaining kaiser halves, pressing gently but firmly.