Cook the potatoes: Halve or quarter the potatoes (depending on size). Boil in well-salted water until they’re almost tender. No need to peel.
Prep the mix-ins: Drain the tuna, pit and tear up the olives, halve the tomatoes, and cut the green beans (or asparagus, or another green veg) into chunky pieces.
Optional - cook eggs: If using eggs, gently lower them into the pot with the potatoes for the last 7 minutes of cooking.
Make the dressing: In a mixing bowl, mash the anchovy with a fork, and mince or grate in the garlic. Add Dijon, red wine vinegar, ½ teaspoon salt, a few cracks of pepper, and olive oil. Mix until mostly smooth.
Cook the green beans: Add the green beans to the potato pot for the last 2 minutes of cooking time so they steam on top of the potatoes. They should still be bright green and crisp-tender.
Drain and cool: Drain the potatoes and beans (and eggs, if using). Let everything cool until you they’re no longer steaming. For the potatoes and beans, spread them out on a baking sheet so they cool quickly. For the eggs, run them under cold water for a minute or so to stop the cooking.
Dress the salad: Toss the potatoes, beans, tomatoes, olives, and tuna with the dressing until everything is coated.
Adjust seasoning: Taste, then add more salt and pepper if needed.
Finish (optional): Top with a soft-boiled egg (or a jammy egg).