Fill a large pot with cold water and add 1 tablespoon kosher salt. Add the whole potatoes and bring to a boil. Meanwhile, working in batches slice off the stem ends of the green beans.
Right when the water comes to a boil, start a timer for 5 minutes and place the green beans in the pot with the potatoes. 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.
Keep cooking the potatoes for about 3 minutes until fork tender (taste test to check). While they are cooking, mince 2 tablespoons 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, 1/2 teaspoon kosher salt, and 1/4 cup warm water. Let stand for 5 minutes, gently stirring occasionally. Then add the capers, parsley (optional) 1 tablespoon olive oil, and a few grinds black pepper. Taste and add additional salt if necessary.
Make the dressing: Make the Lemon Vinaigrette.
Slice the tomatoes into wedges (or cherry tomatoes in half) and sprinkle with a little kosher salt.
Drain the tuna and flake it using a fork, then add a few pinches of kosher salt and a drizzle of olive oil.
To serve, in large shallow bowls or on large platters, add Red Baby Butter Lettuce leaves. Top with green beans, hard-boiled eggs, tomato, tuna, Kalamata olives, and potatoes. Drizzle with Lemon Vinaigrette and serve.