Fill two pots with water - one large one for the potatoes and green beans; one small one for the eggs. Place them both over high heat and bring them to a boil.
Whisk together the shallots, chives, vinegar, mustard, honey, and salt. Add the olive oil while whisking. (Or combine all of the ingredients in a jar and shake well.)
When the small pot of water is boiling, reduce the heat under the small pot of boiling water to a low simmer. Use a spoon to gently lower the eggs into the simmering water. Set a timer for 8 minutes. After 8 minutes, drain the eggs well and run them under cool water.
Meanwhile, season the large pot of boiling water well with salt (for a large pot of salt, you’ll want about 2 tablespoons of kosher salt - it will seem like a lot but that’s how much you need to give the potatoes flavor). Add the potatoes and simmer until they’re nearly tender, about 15 minutes. Add green beans and simmer them both together until tender (this should take 4 to 5 minutes more). Drain well.
Spread the mixed greens out on a large serving platter or in a large salad bowl.
Slice the potatoes in half and toss them with 2 Tablespoons of the vinaigrette. Taste the potatoes and add more salt if needed (potatoes need to be seasoned well for the best flavor). Add them to the salad along with the green beans, tuna, olives, and tomatoes. Slice the eggs in half and add them on top. Season the eggs with some salt and pepper.
Just before serving, drizzle the salad with vinaigrette. Scatter capers over the entire dish.