Heat the oven to 350º F. Put the beets in a single layer in a baking dish. Add water to come about 1/2 inch up the side of the dish. Cover with foil and
roast until the beets can be pierced with a sharp knife, about 45 minutes. When cool, peel and cut them into 1/2-inch wedges, and toss with 1 Tbs. of the vinegar and salt to taste; let sit for 10 to 15 minutes.
Fill a large bowl with cold water. Squeeze the juice of the lemon halves into it. Trim off the top quarter of the artichokes and snap off the tough outer leaves. Using a small, sharp knife, peel the stem and the base of the artichokes, then cut them in half and scoop out the choke with a spoon. As you finish trimming each artichoke, drop it into the bowl of water and lemon juice to prevent browning.
Just before cooking, drain the artichokes well. Warm a medium skillet over medium heat. Add 1⁄4 cup olive oil, the artichokes, wine, 1⁄2 cup water, the thyme, and a pinch of red pepper flakes. Season with salt and simmer, uncovered and stirring occasionally, until the artichokes are tender when pierced at the base with a small, sharp knife, 10 to 15 minutes, depending on the size of the artichokes. If the liquid evaporates before the artichokes are tender, add a splash more water. Let cool at room temperature and then taste for salt.
Put the potatoes in a shallow baking dish or pan just large enough to hold them in a single layer. Drizzle with 1 Tbs. oil, season with a generous amount of salt, and toss well. Add a splash of water (just enough to create a little steam as the potatoes cook). Cover tightly with aluminum
foil and bake until the potatoes can be easily pierced with a small, sharp knife, 30 to 40 minutes, depending on the size of the potatoes. Remove from the oven, vent the foil, and let cool at room temperature.
To make the vinaigrette, combine the garlic, mustard, 1-1/2 Tbs. vinegar, and 1 Tbs. lemon juice in a small bowl. Let sit for 5 to 10 minutes. Whisk in the egg yolk, and then slowly whisk in 3⁄4 cup oil. Thin the vinaigrette with a few drops of cool water if necessary (you want it thin enough to drizzle nicely.) Taste and adjust with more salt or vinegar if necessary. Set aside.
Shortly before serving, halve or quarter the potatoes (depending on size and preference) and season with about 1-1⁄2 Tbs. of the vinaigrette. Set aside.
Season the tuna on both sides with salt and freshly ground black pepper. Warm a large skillet, preferably cast iron, over high heat until very hot. Add 3 Tbs. oil and place the tuna in the skillet. Cook, without moving, until seared and nicely browned, about 2 to 3 minutes, depending on the thickness of the tuna. Turn and cook on the opposite side for another 2 to 3 minutes. (The tuna should be pink in the center.) Transfer to a plate and set aside.
Drain any liquid from the artichokes and discard the thyme. Cut the eggs into quarters and season with salt and freshly ground black pepper. Scatter about half of the arugula around a large platter or individual plates. Tuck the potatoes, artichokes, beets, and eggs in and around the greens. Using your hands, break the tuna into rustic pieces, or slice it with a knife and nestle it in and around the other ingredients. Drizzle about 1⁄4 cup vinaigrette over the platter, or about a Tbs. over individual portions, and serve immediately, passing the remaining vinaigrette at the table.
You can cook the new potatoes, artichokes, beets, and eggs up to a day in advance and refrigerate, but wait to make the vinaigrette shortly before serving. If you refrigerate any element, be sure to let it come to room temperature before finishing the salad.