Boil the potatoes: Fill a large pot with cold water and add 1 tablespoon kosher salt. Add the whole potatoes and bring to a boil. Once boiling, boil for 8 to 12 minutes until fork-tender.
Blanch the beans: Cut the ends off the beans. Bring a separate medium pot of water to a boil with 1/2 tablespoon salt. Boil the beans for about 5 minutes until tender but still bright green. Prepare a large bowl of ice water (or use the same one as for the hard-boiled eggs). Right when the beans are tender, remove them from the boiling water with tongs and transfer them to the ice bath. Remove the beans and dry them with a towel, then mix them with a few pinches of salt and fresh ground pepper.
Finish the potatoes: While the potatoes are cooking, mince the shallot. When the potatoes are done, drain them. When they are cool enough to handle, slice them into bite-sized pieces. Place them in a bowl and gently mix in the minced shallot, white wine vinegar, ½ teaspoon kosher salt, and ¼ cup warm water. Let stand for 5 minutes, gently stirring occasionally. Then add the capers, chopped parsley, 1 tablespoon olive oil, and a few grinds black pepper. Taste and add additional salt if necessary.
Make the dressing: In a medium bowl, whisk together the lemon juice, Dijon mustard, minced garlic, and 1/4 teaspoon kosher salt. Gradually whisk in 6 tablespoons olive oil, working one tablespoon at a time, until creamy and emulsified.
Prep the salad ingredients:Slice the tomatoes into wedges (or cherry tomatoes in half) and sprinkle with a little kosher salt.
Prep the tuna: Drain the tuna and flake it using a fork. If not packed in oil, add a drizzle of olive oil and a few pinches of salt.
Serve: To serve, in large shallow bowls or on large platters, add the Bibb lettuce leaves. Top with the potatoes, green beans, hard-boiled eggs, tomatoes, tuna, and olives. Drizzle with the dressing and serve.