In a medium bowl, whisk together olive oil, lemon juice, mustard, shallot, anchovy paste, and capers. Set aside.
Prepare an ice-water bath in a large bowl, and set aside. Place quail eggs in a medium saucepan, and cover with several inches of cold water. Place the pan, covered, over high heat, stirring occasionally, until just before the water comes to a boil. Uncover, reduce to a simmer, and cook for 2 minutes, stirring eggs gently several times to ensure even cooking. Immediately transfer the eggs, with a slotted spoon, to the ice-water bath. Let cool completely, about 5 minutes. Pat one egg dry at a time; peel, and halve. Set aside.
Fill a medium saucepan with salted water, and bring to a boil. Cook haricots verts until tender, 2 to 3 minutes. Using a slotted spoon, immediately transfer beans to the ice-water bath. Drain well, and pat dry. Drizzle with rice-wine vinegar, and set aside.
Heat a well-seasoned grill pan over high heat. Meanwhile, place potatoes in a medium-size pot. Cover with cold, salted water. Bring to a boil, then reduce heat to simmer. Cook until fork-tender, about 8 minutes. Using a slotted spoon, immediately transfer potatoes to the ice-water bath. Drain well, and pat dry. Cut in half lengthwise, drizzle with vermouth, and reserve.
Lightly brush tuna with olive oil, and season with salt and pepper. Place on grill pan, and cook to rare, 1 or 2 minutes per side.
Toss lettuces with some of the anchovy-caper dressing, and place on large serving platter. In a separate bowl, toss potatoes, haricots verts, and tomatoes with more dressing, and arrange over lettuce.
Slice tuna very thinly on the diagonal, and arrange over salad. Drizzle tuna with dressing. Garnish salad with quail eggs and nicoise olives.