Heat a well-seasoned grill pan over high heat.
In a medium bowl, whisk together olive oil, lemon juice, mustard, shallot, anchovy paste, and capers. Season with salt and pepper. Set aside.
Prepare an ice bath in a large bowl. Set aside. Place eggs in a medium saucepan and cover with several inches of cold water. Place the pan, uncovered, over high heat, stirring occasionally, until just before the water comes to a boil. Cover, turn off the heat, and cook for 12 minutes. Immediately transfer the eggs with a slotted spoon to ice bath. Let cool completely, about 5 minutes. Pat dry, one egg at a time; peel and halve. Set aside.
Prepare an ice bath. Fill a medium saucepan with salted water and bring to a boil. Cook haricots verts until tender, 2 to 3 minutes. Immediately transfer beans, with slotted spoon, to ice bath. Drain well and pat dry. Drizzle with rice wine vinegar. Season with salt and pepper. Set aside.
Prepare an ice bath. Place potatoes in medium-sized pot. Cover with cold salted water. Bring to a boil and reduce heat to simmer. Cook until fork-tender, about 8 minutes. Immediately transfer potatoes with slotted spoon to ice bath. Drain well and pat dry. Cut in half lengthwise. Drizzle with vermouth. Season with salt and pepper. Set aside.
Toss lettuces with some of the anchovy-caper dressing and place on large serving platter. Toss potatoes, haricots verts, and cherry tomatoes separately with dressing and arrange over lettuce.
Arrange tuna over salad and drizzle with dressing. Garnish salad with eggs and Nicoise olives.