Whisk together the lemon juice, minced garlic, mustard, salt, and black pepper in a small bowl until the salt dissolves and the mustard is fully incorporated. Slowly drizzle in the olive oil while whisking constantly to create an emulsified dressing that will coat all the salad components evenly. Set aside at room temperature—this allows the flavors to meld together while you prepare the other elements.
Bring two separate pots of salted water to a boil. Add the halved or quartered potatoes to one pot and cook for 12-15 minutes until just tender when pierced with a fork—don't overcook or they'll become mushy. In the other pot, add the trimmed green beans and cook for 4-5 minutes until bright green and tender-crisp. Drain the potatoes and green beans separately, then spread them on a clean kitchen towel or paper towels to cool and dry completely. I like to let them cool to room temperature before dressing so they absorb the vinaigrette better without becoming soggy.
While the potatoes and beans cook, bring a small pot of water to a rolling boil and carefully place the eggs in it. Cook for 10 minutes for fully hard-boiled eggs with no trace of runniness. Transfer to an ice bath immediately and let them cool for at least 5 minutes, then peel and slice each egg into quarters. Meanwhile, slice the tomatoes into ½-inch wedges, roughly chop the flat-leaf parsley, and prepare the lettuce by washing and separating the leaves. I keep the lettuce leaves whole or torn into large pieces so they provide structure to the salad rather than turning into mush.
Lay the cos lettuce leaves on a large serving platter or individual plates to create a base. Arrange the cooled potatoes and green beans from Step 2 in separate sections around the plate, then add the tomato wedges in another section. Nestle the hard-boiled egg quarters, Nicoise olives, and drained tuna into their own clusters on the plate, creating distinct sections of color and texture. Scatter the capers over the salad and sprinkle with fresh parsley. This traditional arrangement makes the salad visually appealing and allows each component to shine individually.
Pour the vinaigrette from Step 1 over the assembled salad just before serving, or drizzle it over individual plates if preferred. Toss gently if desired, or let diners dress their own portions to their taste. Serve immediately while the potatoes and beans are still cool and the lettuce is crisp.